Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Kinetic models of economy. The conservative case
Giuseppe Toscani
Department of Mathematics
University of Pavia, Italy
Orleans, March 15 2007
KTASEEM
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Outline
1
Formation of Pareto tails
Introduction
Other systems with overpopulated tails
Market and trades
2
Kinetic models
The Boltzmann equation
Formation of tails
Examples
3
Conclusions
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Outline
1
Formation of Pareto tails
Introduction
Other systems with overpopulated tails
Market and trades
2
Kinetic models
The Boltzmann equation
Formation of tails
Examples
3
Conclusions
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Outline
1
Formation of Pareto tails
Introduction
Other systems with overpopulated tails
Market and trades
2
Kinetic models
The Boltzmann equation
Formation of tails
Examples
3
Conclusions
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
1
Formation of Pareto tails
Introduction
Other systems with overpopulated tails
Market and trades
2
Kinetic models
The Boltzmann equation
Formation of tails
Examples
3
Conclusions
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The goal
Introduce kinetic models of Boltzmann type to study the
evolution of the distribution of wealth in a simple market
economy.
In most of these models, the market is represented as an ideal
gas, where each molecule is identified with an agent, and each
trading event between two agents is considered to be an
elastic – or rather money conserving – collision between two
molecules.
The founding idea is that a trading market composed of a
sufficiently large number of agents can be described using the
laws of statistical mechanics as it happens in a physical
system composed of many interacting particles.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The goal
Introduce kinetic models of Boltzmann type to study the
evolution of the distribution of wealth in a simple market
economy.
In most of these models, the market is represented as an ideal
gas, where each molecule is identified with an agent, and each
trading event between two agents is considered to be an
elastic – or rather money conserving – collision between two
molecules.
The founding idea is that a trading market composed of a
sufficiently large number of agents can be described using the
laws of statistical mechanics as it happens in a physical
system composed of many interacting particles.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The goal
Introduce kinetic models of Boltzmann type to study the
evolution of the distribution of wealth in a simple market
economy.
In most of these models, the market is represented as an ideal
gas, where each molecule is identified with an agent, and each
trading event between two agents is considered to be an
elastic – or rather money conserving – collision between two
molecules.
The founding idea is that a trading market composed of a
sufficiently large number of agents can be described using the
laws of statistical mechanics as it happens in a physical
system composed of many interacting particles.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states
The knowledge of the large-wealth behavior of the steady
state density is of primary importance since it determines a
posteriori if the model fits data of real economies.
More than a hundred years ago, the Italian economist Vilfredo
Pareto [V.Pareto, Cours d’Economie Politique, (1897)] first
quantified the large-wealth behavior of the income distribution
in a society and found it to follow a power-law distribution.
If f (w ) is the probability density function of agents with
wealth w , and w is sufficiently large,
Z
∞
F (w ) =
f (w∗ ) dw∗ ∼ w −µ .
w
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states
The knowledge of the large-wealth behavior of the steady
state density is of primary importance since it determines a
posteriori if the model fits data of real economies.
More than a hundred years ago, the Italian economist Vilfredo
Pareto [V.Pareto, Cours d’Economie Politique, (1897)] first
quantified the large-wealth behavior of the income distribution
in a society and found it to follow a power-law distribution.
If f (w ) is the probability density function of agents with
wealth w , and w is sufficiently large,
Z
∞
F (w ) =
f (w∗ ) dw∗ ∼ w −µ .
w
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states
The knowledge of the large-wealth behavior of the steady
state density is of primary importance since it determines a
posteriori if the model fits data of real economies.
More than a hundred years ago, the Italian economist Vilfredo
Pareto [V.Pareto, Cours d’Economie Politique, (1897)] first
quantified the large-wealth behavior of the income distribution
in a society and found it to follow a power-law distribution.
If f (w ) is the probability density function of agents with
wealth w , and w is sufficiently large,
Z
∞
F (w ) =
f (w∗ ) dw∗ ∼ w −µ .
w
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states II
Pareto mistakenly believed that power laws apply to the whole
distribution with a universal exponent µ approximatively equal
to 1.5.
Later, Mandelbrot proposed a weak Pareto law that applies
only to high incomes [B.Mandelbrot (1960)].
The exponent µ, usually called the Pareto index, was initially
found to have a value between 1 and 3 .
Considerable investigations with real data during the last ten
years revealed that the tail of the income distribution indeed
follows the above mentioned behavior and the value of the
Pareto index is generally seen to vary between 1 and 2.5 (USA
∼ 1.6, Japan ∼ 1.8 − 2.2)
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)]
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states II
Pareto mistakenly believed that power laws apply to the whole
distribution with a universal exponent µ approximatively equal
to 1.5.
Later, Mandelbrot proposed a weak Pareto law that applies
only to high incomes [B.Mandelbrot (1960)].
The exponent µ, usually called the Pareto index, was initially
found to have a value between 1 and 3 .
Considerable investigations with real data during the last ten
years revealed that the tail of the income distribution indeed
follows the above mentioned behavior and the value of the
Pareto index is generally seen to vary between 1 and 2.5 (USA
∼ 1.6, Japan ∼ 1.8 − 2.2)
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)]
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states II
Pareto mistakenly believed that power laws apply to the whole
distribution with a universal exponent µ approximatively equal
to 1.5.
Later, Mandelbrot proposed a weak Pareto law that applies
only to high incomes [B.Mandelbrot (1960)].
The exponent µ, usually called the Pareto index, was initially
found to have a value between 1 and 3 .
Considerable investigations with real data during the last ten
years revealed that the tail of the income distribution indeed
follows the above mentioned behavior and the value of the
Pareto index is generally seen to vary between 1 and 2.5 (USA
∼ 1.6, Japan ∼ 1.8 − 2.2)
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)]
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The steady states II
Pareto mistakenly believed that power laws apply to the whole
distribution with a universal exponent µ approximatively equal
to 1.5.
Later, Mandelbrot proposed a weak Pareto law that applies
only to high incomes [B.Mandelbrot (1960)].
The exponent µ, usually called the Pareto index, was initially
found to have a value between 1 and 3 .
Considerable investigations with real data during the last ten
years revealed that the tail of the income distribution indeed
follows the above mentioned behavior and the value of the
Pareto index is generally seen to vary between 1 and 2.5 (USA
∼ 1.6, Japan ∼ 1.8 − 2.2)
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)]
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Consequences
The main consequence is that typically less than 10% of the
population in any country possesses about 40% of the total
wealth of that country and they follow the above law.
The rest of the low income population, in fact the majority
(90% or more), follow a different distribution which is debated
to be either Gibbs
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)] or log-normal
[T. Di Matteo, T. Aste, S.T. Hyde (2004)].
It is possible to reproduce these results by means of a kinetic
approach?
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Consequences
The main consequence is that typically less than 10% of the
population in any country possesses about 40% of the total
wealth of that country and they follow the above law.
The rest of the low income population, in fact the majority
(90% or more), follow a different distribution which is debated
to be either Gibbs
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)] or log-normal
[T. Di Matteo, T. Aste, S.T. Hyde (2004)].
It is possible to reproduce these results by means of a kinetic
approach?
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Consequences
The main consequence is that typically less than 10% of the
population in any country possesses about 40% of the total
wealth of that country and they follow the above law.
The rest of the low income population, in fact the majority
(90% or more), follow a different distribution which is debated
to be either Gibbs
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)] or log-normal
[T. Di Matteo, T. Aste, S.T. Hyde (2004)].
It is possible to reproduce these results by means of a kinetic
approach?
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A typical picture
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The cooling of a granular gas
A well–known phenomenon in the large–time behavior of the
Boltzmann equation with dissipative interactions is the
formation of overpopulated tails
[Bobylev, A. V.; Gamba, I. M.; Panferov, V. A. (2004)].
Exact results on the behavior of these tails have been obtained
for simplified models, in particular for a gas inelastic Maxwell
particles.
In one dimension
f (w ) =
2
π(1 + v 2 )2
[Baldassarri A., Marini Bettolo Marconi U., Puglisi A. (2002)]
Analogous results (with no explicit self-similar solution) hold in
higher dimensions [Bobylev A.V., Cercignani C. (2003)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The cooling of a granular gas
A well–known phenomenon in the large–time behavior of the
Boltzmann equation with dissipative interactions is the
formation of overpopulated tails
[Bobylev, A. V.; Gamba, I. M.; Panferov, V. A. (2004)].
Exact results on the behavior of these tails have been obtained
for simplified models, in particular for a gas inelastic Maxwell
particles.
In one dimension
f (w ) =
2
π(1 + v 2 )2
[Baldassarri A., Marini Bettolo Marconi U., Puglisi A. (2002)]
Analogous results (with no explicit self-similar solution) hold in
higher dimensions [Bobylev A.V., Cercignani C. (2003)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
The cooling of a granular gas
A well–known phenomenon in the large–time behavior of the
Boltzmann equation with dissipative interactions is the
formation of overpopulated tails
[Bobylev, A. V.; Gamba, I. M.; Panferov, V. A. (2004)].
Exact results on the behavior of these tails have been obtained
for simplified models, in particular for a gas inelastic Maxwell
particles.
In one dimension
f (w ) =
2
π(1 + v 2 )2
[Baldassarri A., Marini Bettolo Marconi U., Puglisi A. (2002)]
Analogous results (with no explicit self-similar solution) hold in
higher dimensions [Bobylev A.V., Cercignani C. (2003)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Fast diffusion equations
A further example of a PDE with a self-similar solution with
overpopulated tails is represented by fast diffusion equations.
The logarithmic diffusion [J.A. Carrillo, J.L. Vázquez (2003)].
∂f
∂ 2 log f
=
,
∂t
∂w 2
t≥0
has the Cauchy distribution as self-similar solution
fs (w ) =
1
π(1 + v 2 )
The connection between the fast diffusion equation and
classical kinetic theory relies in a result of McKean on the
central limit theorem for Carleman model
[H.P. McKean (1975)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Fast diffusion equations
A further example of a PDE with a self-similar solution with
overpopulated tails is represented by fast diffusion equations.
The logarithmic diffusion [J.A. Carrillo, J.L. Vázquez (2003)].
∂f
∂ 2 log f
=
,
∂t
∂w 2
t≥0
has the Cauchy distribution as self-similar solution
fs (w ) =
1
π(1 + v 2 )
The connection between the fast diffusion equation and
classical kinetic theory relies in a result of McKean on the
central limit theorem for Carleman model
[H.P. McKean (1975)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Fast diffusion equations
A further example of a PDE with a self-similar solution with
overpopulated tails is represented by fast diffusion equations.
The logarithmic diffusion [J.A. Carrillo, J.L. Vázquez (2003)].
∂f
∂ 2 log f
=
,
∂t
∂w 2
t≥0
has the Cauchy distribution as self-similar solution
fs (w ) =
1
π(1 + v 2 )
The connection between the fast diffusion equation and
classical kinetic theory relies in a result of McKean on the
central limit theorem for Carleman model
[H.P. McKean (1975)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Fast diffusion equations
A further example of a PDE with a self-similar solution with
overpopulated tails is represented by fast diffusion equations.
The logarithmic diffusion [J.A. Carrillo, J.L. Vázquez (2003)].
∂f
∂ 2 log f
=
,
∂t
∂w 2
t≥0
has the Cauchy distribution as self-similar solution
fs (w ) =
1
π(1 + v 2 )
The connection between the fast diffusion equation and
classical kinetic theory relies in a result of McKean on the
central limit theorem for Carleman model
[H.P. McKean (1975)].
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Common features
Economic and dissipative systems have much in common.
Pareto tails
In a strong economy the mean
wealth m(t) is increasing
(only mass conservation).
A time-independent density
profile is obtained with the
normalized density
fn (w ) = m(t)f (m(t)w , t)
of unit mean wealth.
Giuseppe Toscani
Granular gases
In a gas with dissipative
interactions the temperature
E (t) is decreasing (mass and
momentum are conserved).
In this case one refers to a
normalized density
p
p
fn (w ) = E (t)f ( E (t)w , t)
of unit energy.
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Common features
Economic and dissipative systems have much in common.
Pareto tails
In a strong economy the mean
wealth m(t) is increasing
(only mass conservation).
A time-independent density
profile is obtained with the
normalized density
fn (w ) = m(t)f (m(t)w , t)
of unit mean wealth.
Giuseppe Toscani
Granular gases
In a gas with dissipative
interactions the temperature
E (t) is decreasing (mass and
momentum are conserved).
In this case one refers to a
normalized density
p
p
fn (w ) = E (t)f ( E (t)w , t)
of unit energy.
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Common features
Economic and dissipative systems have much in common.
Pareto tails
In a strong economy the mean
wealth m(t) is increasing
(only mass conservation).
A time-independent density
profile is obtained with the
normalized density
fn (w ) = m(t)f (m(t)w , t)
of unit mean wealth.
Giuseppe Toscani
Granular gases
In a gas with dissipative
interactions the temperature
E (t) is decreasing (mass and
momentum are conserved).
In this case one refers to a
normalized density
p
p
fn (w ) = E (t)f ( E (t)w , t)
of unit energy.
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Common features
Economic and dissipative systems have much in common.
Pareto tails
In a strong economy the mean
wealth m(t) is increasing
(only mass conservation).
A time-independent density
profile is obtained with the
normalized density
fn (w ) = m(t)f (m(t)w , t)
of unit mean wealth.
Giuseppe Toscani
Granular gases
In a gas with dissipative
interactions the temperature
E (t) is decreasing (mass and
momentum are conserved).
In this case one refers to a
normalized density
p
p
fn (w ) = E (t)f ( E (t)w , t)
of unit energy.
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Common features
Economic and dissipative systems have much in common.
Pareto tails
In a strong economy the mean
wealth m(t) is increasing
(only mass conservation).
A time-independent density
profile is obtained with the
normalized density
fn (w ) = m(t)f (m(t)w , t)
of unit mean wealth.
Giuseppe Toscani
Granular gases
In a gas with dissipative
interactions the temperature
E (t) is decreasing (mass and
momentum are conserved).
In this case one refers to a
normalized density
p
p
fn (w ) = E (t)f ( E (t)w , t)
of unit energy.
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A reasonable conjecture
It can be reasonably conjectured that the formation of
overpopulated tails depends mainly on two facts:
The kinetic system is based on binary collisions that generate
convolution-like collisional operators.
The collision is such that some conservation law in the kinetic
system is missed at a microscopic level.
The classical elastic Boltzmann equation is pointwise
conservative, and the steady state is a gaussian function (the
Maxwellian)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A reasonable conjecture
It can be reasonably conjectured that the formation of
overpopulated tails depends mainly on two facts:
The kinetic system is based on binary collisions that generate
convolution-like collisional operators.
The collision is such that some conservation law in the kinetic
system is missed at a microscopic level.
The classical elastic Boltzmann equation is pointwise
conservative, and the steady state is a gaussian function (the
Maxwellian)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A reasonable conjecture
It can be reasonably conjectured that the formation of
overpopulated tails depends mainly on two facts:
The kinetic system is based on binary collisions that generate
convolution-like collisional operators.
The collision is such that some conservation law in the kinetic
system is missed at a microscopic level.
The classical elastic Boltzmann equation is pointwise
conservative, and the steady state is a gaussian function (the
Maxwellian)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A reasonable conjecture
It can be reasonably conjectured that the formation of
overpopulated tails depends mainly on two facts:
The kinetic system is based on binary collisions that generate
convolution-like collisional operators.
The collision is such that some conservation law in the kinetic
system is missed at a microscopic level.
The classical elastic Boltzmann equation is pointwise
conservative, and the steady state is a gaussian function (the
Maxwellian)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades I
The goal of a kinetic model of a simple market economy, is to
describe the evolution of the distribution of wealth by means
of microscopic interactions among agents or individuals which
perform exchange of money.
Each trade can be interpreted as an interaction where a fraction of
the money changes hands. One generally assumes that this wealth
after the interaction is non-negative, which corresponds to impose
that no debts are allowed.
From a microscopic view point, we will consider in the
following binary interactions described by the rules
v ∗ = p1 v + q1 w ;
Giuseppe Toscani
w ∗ = p2 v + q2 w .
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades I
The goal of a kinetic model of a simple market economy, is to
describe the evolution of the distribution of wealth by means
of microscopic interactions among agents or individuals which
perform exchange of money.
Each trade can be interpreted as an interaction where a fraction of
the money changes hands. One generally assumes that this wealth
after the interaction is non-negative, which corresponds to impose
that no debts are allowed.
From a microscopic view point, we will consider in the
following binary interactions described by the rules
v ∗ = p1 v + q1 w ;
Giuseppe Toscani
w ∗ = p2 v + q2 w .
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades I
The goal of a kinetic model of a simple market economy, is to
describe the evolution of the distribution of wealth by means
of microscopic interactions among agents or individuals which
perform exchange of money.
Each trade can be interpreted as an interaction where a fraction of
the money changes hands. One generally assumes that this wealth
after the interaction is non-negative, which corresponds to impose
that no debts are allowed.
From a microscopic view point, we will consider in the
following binary interactions described by the rules
v ∗ = p1 v + q1 w ;
Giuseppe Toscani
w ∗ = p2 v + q2 w .
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades II
(v , w ) denote the (positive) money of two arbitrary individuals
before the trade and (v ∗ , w ∗ ) the money after the trade.
We will not allow agents to have debts, and thus the
interaction takes place only if v ∗ ≥ 0 and w ∗ ≥ 0.
The transaction coefficients pi , qi , i = 1, 2 can be either given
constant or random quantities, with the obvious constraint to
be nonnegative.
The total amount v + w of wealth involved in the trade
changes into
v ∗ + w ∗ = (p1 + p2 )v + (q1 + q2 )w ,
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades II
(v , w ) denote the (positive) money of two arbitrary individuals
before the trade and (v ∗ , w ∗ ) the money after the trade.
We will not allow agents to have debts, and thus the
interaction takes place only if v ∗ ≥ 0 and w ∗ ≥ 0.
The transaction coefficients pi , qi , i = 1, 2 can be either given
constant or random quantities, with the obvious constraint to
be nonnegative.
The total amount v + w of wealth involved in the trade
changes into
v ∗ + w ∗ = (p1 + p2 )v + (q1 + q2 )w ,
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades II
(v , w ) denote the (positive) money of two arbitrary individuals
before the trade and (v ∗ , w ∗ ) the money after the trade.
We will not allow agents to have debts, and thus the
interaction takes place only if v ∗ ≥ 0 and w ∗ ≥ 0.
The transaction coefficients pi , qi , i = 1, 2 can be either given
constant or random quantities, with the obvious constraint to
be nonnegative.
The total amount v + w of wealth involved in the trade
changes into
v ∗ + w ∗ = (p1 + p2 )v + (q1 + q2 )w ,
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades II
(v , w ) denote the (positive) money of two arbitrary individuals
before the trade and (v ∗ , w ∗ ) the money after the trade.
We will not allow agents to have debts, and thus the
interaction takes place only if v ∗ ≥ 0 and w ∗ ≥ 0.
The transaction coefficients pi , qi , i = 1, 2 can be either given
constant or random quantities, with the obvious constraint to
be nonnegative.
The total amount v + w of wealth involved in the trade
changes into
v ∗ + w ∗ = (p1 + p2 )v + (q1 + q2 )w ,
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades III
Unless both
p1 + p2 = 1,
q1 + q2 = 1,
the total wealth put into the trade is not conserved.
Condition relaxed in presence of random coefficients. In this
case, the total wealth put into the trade, even if not pointwise
conserved, can be conserved in the mean
hp1 + p2 i = 1,
hq1 + q2 i = 1
These conditions define conservative trades. These
interactions are such that the mean wealth in the kinetic
model is a conserved quantity.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades III
Unless both
p1 + p2 = 1,
q1 + q2 = 1,
the total wealth put into the trade is not conserved.
Condition relaxed in presence of random coefficients. In this
case, the total wealth put into the trade, even if not pointwise
conserved, can be conserved in the mean
hp1 + p2 i = 1,
hq1 + q2 i = 1
These conditions define conservative trades. These
interactions are such that the mean wealth in the kinetic
model is a conserved quantity.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Trades III
Unless both
p1 + p2 = 1,
q1 + q2 = 1,
the total wealth put into the trade is not conserved.
Condition relaxed in presence of random coefficients. In this
case, the total wealth put into the trade, even if not pointwise
conserved, can be conserved in the mean
hp1 + p2 i = 1,
hq1 + q2 i = 1
These conditions define conservative trades. These
interactions are such that the mean wealth in the kinetic
model is a conserved quantity.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chakraborty - Chakrabarthi model
In [A. Chakraborti, B.K. Chakrabarti (2000)] the agents
taking part in trading change their money according to the
rule
v ∗ = v + ∆(v , w );
w ∗ = w − ∆(v , w ),
where ∆(v , w ) represents the amount of money to be
exchanged.
The realization of ∆(v , w ) takes into account the basic fact
that the agents always keep some money in hand when
trading.
The ratio of saving to all of the money held (usually denoted
by λ or s) is called the saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chakraborty - Chakrabarthi model
In [A. Chakraborti, B.K. Chakrabarti (2000)] the agents
taking part in trading change their money according to the
rule
v ∗ = v + ∆(v , w );
w ∗ = w − ∆(v , w ),
where ∆(v , w ) represents the amount of money to be
exchanged.
The realization of ∆(v , w ) takes into account the basic fact
that the agents always keep some money in hand when
trading.
The ratio of saving to all of the money held (usually denoted
by λ or s) is called the saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chakraborty - Chakrabarthi model
In [A. Chakraborti, B.K. Chakrabarti (2000)] the agents
taking part in trading change their money according to the
rule
v ∗ = v + ∆(v , w );
w ∗ = w − ∆(v , w ),
where ∆(v , w ) represents the amount of money to be
exchanged.
The realization of ∆(v , w ) takes into account the basic fact
that the agents always keep some money in hand when
trading.
The ratio of saving to all of the money held (usually denoted
by λ or s) is called the saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
C-C model II
The amount of money to be exchanged is
∆(v , w ) = (1 − λ) [( − 1)v + w ] ,
0 ≤ ≤ 1 is a random fraction, or even = 12 .
Mixing parameters (λ saving rate)
p1 = 1 − (1 − λ)(1 − ),
p2 = (1 − λ)(1 − ),
q1 = (1 − λ)
q2 = 1 − (1 − λ).
The trade is pointwise conservative.
Giuseppe Toscani
Kinetic models of economy
(1)
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
C-C model II
The amount of money to be exchanged is
∆(v , w ) = (1 − λ) [( − 1)v + w ] ,
0 ≤ ≤ 1 is a random fraction, or even = 12 .
Mixing parameters (λ saving rate)
p1 = 1 − (1 − λ)(1 − ),
p2 = (1 − λ)(1 − ),
q1 = (1 − λ)
q2 = 1 − (1 − λ).
The trade is pointwise conservative.
Giuseppe Toscani
Kinetic models of economy
(1)
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
C-C model II
The amount of money to be exchanged is
∆(v , w ) = (1 − λ) [( − 1)v + w ] ,
0 ≤ ≤ 1 is a random fraction, or even = 12 .
Mixing parameters (λ saving rate)
p1 = 1 − (1 − λ)(1 − ),
p2 = (1 − λ)(1 − ),
q1 = (1 − λ)
q2 = 1 − (1 − λ).
The trade is pointwise conservative.
Giuseppe Toscani
Kinetic models of economy
(1)
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chatterjee - Chakrabarti - Manna model
In [A. Chatterjee, B.K. Chakrabarti, S.S. Manna (2004)]
Chakrabarthi research group introduced random saving rate
for the agents
p1 = 1 − (1 − λv )(1 − ), q1 = (1 − λw )
p2 = (1 − λv )(1 − ), q2 = 1 − (1 − λw ),
The random quantities λv and λw model the individual saving
rates of agents with wealth v and w , respectively.
Model still pointwise conservative. Agents with high saving
propensity tend to become richer than individuals with lower
saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chatterjee - Chakrabarti - Manna model
In [A. Chatterjee, B.K. Chakrabarti, S.S. Manna (2004)]
Chakrabarthi research group introduced random saving rate
for the agents
p1 = 1 − (1 − λv )(1 − ), q1 = (1 − λw )
p2 = (1 − λv )(1 − ), q2 = 1 − (1 − λw ),
The random quantities λv and λw model the individual saving
rates of agents with wealth v and w , respectively.
Model still pointwise conservative. Agents with high saving
propensity tend to become richer than individuals with lower
saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Chatterjee - Chakrabarti - Manna model
In [A. Chatterjee, B.K. Chakrabarti, S.S. Manna (2004)]
Chakrabarthi research group introduced random saving rate
for the agents
p1 = 1 − (1 − λv )(1 − ), q1 = (1 − λw )
p2 = (1 − λv )(1 − ), q2 = 1 − (1 − λw ),
The random quantities λv and λw model the individual saving
rates of agents with wealth v and w , respectively.
Model still pointwise conservative. Agents with high saving
propensity tend to become richer than individuals with lower
saving rate.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
λ is the constant saving rate, while η and η̃ are independent
centered random variables.
The random term contains the effects of an open economy
describing the risks (or returns) of the market
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
λ is the constant saving rate, while η and η̃ are independent
centered random variables.
The random term contains the effects of an open economy
describing the risks (or returns) of the market
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
λ is the constant saving rate, while η and η̃ are independent
centered random variables.
The random term contains the effects of an open economy
describing the risks (or returns) of the market
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model II
If η, η̃ ≥ −λ, the random coefficients pi , qi , i = 1, 2 are
nonnegative.
p1 = λ + η,
q1 = 1 − λ
p2 = 1 − λ,
q2 = λ + η̃.
(2)
The trade is conservative only in the mean, since
hp1 + p2 i = 1, but p1 + p2 = 1 + η 6= 1 with probability one.
Pointwise conservative (CC and CCM models) and
conservative in the mean trades (CPT model) have different
behaviors!
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model II
If η, η̃ ≥ −λ, the random coefficients pi , qi , i = 1, 2 are
nonnegative.
p1 = λ + η,
q1 = 1 − λ
p2 = 1 − λ,
q2 = λ + η̃.
(2)
The trade is conservative only in the mean, since
hp1 + p2 i = 1, but p1 + p2 = 1 + η 6= 1 with probability one.
Pointwise conservative (CC and CCM models) and
conservative in the mean trades (CPT model) have different
behaviors!
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Cordier - Pareschi - Toscani model II
If η, η̃ ≥ −λ, the random coefficients pi , qi , i = 1, 2 are
nonnegative.
p1 = λ + η,
q1 = 1 − λ
p2 = 1 − λ,
q2 = λ + η̃.
(2)
The trade is conservative only in the mean, since
hp1 + p2 i = 1, but p1 + p2 = 1 + η 6= 1 with probability one.
Pointwise conservative (CC and CCM models) and
conservative in the mean trades (CPT model) have different
behaviors!
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Slanina model
Non-conservative models have been considered by F. Slanina
[F. Slanina (2004)]
p1 = λ,
q1 = 1 − λ + p2 = 1 − λ + ,
q2 = λ.
(3)
is a fixed positive constant, so that the total money put into
the trade increases, and v ∗ + w ∗ = (1 + )(v + w ).
This type of trade intends to introduce (artificially) the effects
of a strong economy, which is such that the total wealth is
increasing in time.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Slanina model
Non-conservative models have been considered by F. Slanina
[F. Slanina (2004)]
p1 = λ,
q1 = 1 − λ + p2 = 1 − λ + ,
q2 = λ.
(3)
is a fixed positive constant, so that the total money put into
the trade increases, and v ∗ + w ∗ = (1 + )(v + w ).
This type of trade intends to introduce (artificially) the effects
of a strong economy, which is such that the total wealth is
increasing in time.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
Slanina model
Non-conservative models have been considered by F. Slanina
[F. Slanina (2004)]
p1 = λ,
q1 = 1 − λ + p2 = 1 − λ + ,
q2 = λ.
(3)
is a fixed positive constant, so that the total money put into
the trade increases, and v ∗ + w ∗ = (1 + )(v + w ).
This type of trade intends to introduce (artificially) the effects
of a strong economy, which is such that the total wealth is
increasing in time.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A remark on non conservative models
A non conservative trade can be obtained in a more natural
way simply allowing the random variables in CPT-trade to
assume values on the whole space, and at the same time
discarding the trades for which the post-trade wealths are non
positive.
Same property for the original transfer model proposed by A.
Drǎgulesku and V. Yakovenko
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)], where ∆(v , w )
is decided by the following trading rule
∆(v , w ) = (v + w ).
2
is a random number, 0 ≤ ≤ 1. If the payer can not afford
the money to be exchanged, the trade does not take place.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A remark on non conservative models
A non conservative trade can be obtained in a more natural
way simply allowing the random variables in CPT-trade to
assume values on the whole space, and at the same time
discarding the trades for which the post-trade wealths are non
positive.
Same property for the original transfer model proposed by A.
Drǎgulesku and V. Yakovenko
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)], where ∆(v , w )
is decided by the following trading rule
∆(v , w ) = (v + w ).
2
is a random number, 0 ≤ ≤ 1. If the payer can not afford
the money to be exchanged, the trade does not take place.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Introduction
Other systems with overpopulated tails
Market and trades
A remark on non conservative models
A non conservative trade can be obtained in a more natural
way simply allowing the random variables in CPT-trade to
assume values on the whole space, and at the same time
discarding the trades for which the post-trade wealths are non
positive.
Same property for the original transfer model proposed by A.
Drǎgulesku and V. Yakovenko
[A.Drǎgulescu, V.M.Yakovenko (2000-2001)], where ∆(v , w )
is decided by the following trading rule
∆(v , w ) = (v + w ).
2
is a random number, 0 ≤ ≤ 1. If the payer can not afford
the money to be exchanged, the trade does not take place.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
1
Formation of Pareto tails
Introduction
Other systems with overpopulated tails
Market and trades
2
Kinetic models
The Boltzmann equation
Formation of tails
Examples
3
Conclusions
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation
f (v , t) denotes the distribution of agents with wealth v ∈ IR +
at time t ≥ 0. A Boltzmann-like equation for f (v , t) can be
derived by standard methods of kinetic theory.
A useful way of writing this equation is to use the so-called
weak form
Z
Z
d
Q(f , f )(v )φ(v ) dv ,
f (v )φ(v ) dv =
dt IR +
IR +
the right-hand side
1
2
Z
Z
∗
∗
(φ(v ) + φ(w ) − φ(v ) − φ(w ))f (v )f (w )dv dw
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
.
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation
f (v , t) denotes the distribution of agents with wealth v ∈ IR +
at time t ≥ 0. A Boltzmann-like equation for f (v , t) can be
derived by standard methods of kinetic theory.
A useful way of writing this equation is to use the so-called
weak form
Z
Z
d
Q(f , f )(v )φ(v ) dv ,
f (v )φ(v ) dv =
dt IR +
IR +
the right-hand side
1
2
Z
Z
∗
∗
(φ(v ) + φ(w ) − φ(v ) − φ(w ))f (v )f (w )dv dw
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
.
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation
f (v , t) denotes the distribution of agents with wealth v ∈ IR +
at time t ≥ 0. A Boltzmann-like equation for f (v , t) can be
derived by standard methods of kinetic theory.
A useful way of writing this equation is to use the so-called
weak form
Z
Z
d
Q(f , f )(v )φ(v ) dv ,
f (v )φ(v ) dv =
dt IR +
IR +
the right-hand side
1
2
Z
Z
∗
∗
(φ(v ) + φ(w ) − φ(v ) − φ(w ))f (v )f (w )dv dw
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
.
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation II
The kinetic equation is the analogous of the Boltzmann
equation for Maxwell molecules, where the collision frequency
is assumed to be constant.
Higher order
R nmoments can be evaluated recursively, since the
integrals v f (v , t) dv obey a closed hierarchy of equations.
Fix the initial density to satisfy
Z
Z
vf0 (v ) dv = M.
f0 (v ) dv = 1 ;
IR +
IR +
Conditions are propagated in time. For all t > 0
Z
Z
f (v , t) dv = 1 ;
vf (v , t) dv = M
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation II
The kinetic equation is the analogous of the Boltzmann
equation for Maxwell molecules, where the collision frequency
is assumed to be constant.
Higher order
R nmoments can be evaluated recursively, since the
integrals v f (v , t) dv obey a closed hierarchy of equations.
Fix the initial density to satisfy
Z
Z
vf0 (v ) dv = M.
f0 (v ) dv = 1 ;
IR +
IR +
Conditions are propagated in time. For all t > 0
Z
Z
f (v , t) dv = 1 ;
vf (v , t) dv = M
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation II
The kinetic equation is the analogous of the Boltzmann
equation for Maxwell molecules, where the collision frequency
is assumed to be constant.
Higher order
R nmoments can be evaluated recursively, since the
integrals v f (v , t) dv obey a closed hierarchy of equations.
Fix the initial density to satisfy
Z
Z
vf0 (v ) dv = M.
f0 (v ) dv = 1 ;
IR +
IR +
Conditions are propagated in time. For all t > 0
Z
Z
f (v , t) dv = 1 ;
vf (v , t) dv = M
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation II
The kinetic equation is the analogous of the Boltzmann
equation for Maxwell molecules, where the collision frequency
is assumed to be constant.
Higher order
R nmoments can be evaluated recursively, since the
integrals v f (v , t) dv obey a closed hierarchy of equations.
Fix the initial density to satisfy
Z
Z
vf0 (v ) dv = M.
f0 (v ) dv = 1 ;
IR +
IR +
Conditions are propagated in time. For all t > 0
Z
Z
f (v , t) dv = 1 ;
vf (v , t) dv = M
IR +
IR +
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation III
The existence of a solution to Boltzmann equation can be
seen easily using the same methods available for the elastic
Kac model.
A Boltzmann-like equation with constant kernel easily studied
by means of Fourier transform [A.V. Bobylev (1988)].
The Fourier version
∂b
f (ξ, t)
b b
=Q
f ,b
f (ξ, t),
∂t
where
E
1 Db
b b
Q
f ,b
f (ξ) =
f (p1 ξ)b
f (q1 ξ) + b
f (p2 ξ)b
f (q2 ξ) − b
f (ξ).
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation III
The existence of a solution to Boltzmann equation can be
seen easily using the same methods available for the elastic
Kac model.
A Boltzmann-like equation with constant kernel easily studied
by means of Fourier transform [A.V. Bobylev (1988)].
The Fourier version
∂b
f (ξ, t)
b b
=Q
f ,b
f (ξ, t),
∂t
where
E
1 Db
b b
Q
f ,b
f (ξ) =
f (p1 ξ)b
f (q1 ξ) + b
f (p2 ξ)b
f (q2 ξ) − b
f (ξ).
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation III
The existence of a solution to Boltzmann equation can be
seen easily using the same methods available for the elastic
Kac model.
A Boltzmann-like equation with constant kernel easily studied
by means of Fourier transform [A.V. Bobylev (1988)].
The Fourier version
∂b
f (ξ, t)
b b
=Q
f ,b
f (ξ, t),
∂t
where
E
1 Db
b b
Q
f ,b
f (ξ) =
f (p1 ξ)b
f (q1 ξ) + b
f (p2 ξ)b
f (q2 ξ) − b
f (ξ).
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation IV
Introduce a metric
ds (f , g ) = sup
ξ∈IR
|b
f (ξ) − gb(ξ)|
|ξ|s
s = m + α, m integer and 0 ≤ α < 1.
Theorem
Let f1 (t) and f2 (t) be two solutions of the Boltzmann equation,
corresponding to initial values with the same mean. Assume that
ds (f1,0 , f2,0 ) is bounded for some positive constant s.
Then, for all times t ≥ 0,
! )
(
2
1X s
ds (f1 (t), f2 (t)) ≤ e
hpi + qis i − 1 t ds (f1,0 , f2,0 ).
2
i=1
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation IV
Introduce a metric
ds (f , g ) = sup
ξ∈IR
|b
f (ξ) − gb(ξ)|
|ξ|s
s = m + α, m integer and 0 ≤ α < 1.
Theorem
Let f1 (t) and f2 (t) be two solutions of the Boltzmann equation,
corresponding to initial values with the same mean. Assume that
ds (f1,0 , f2,0 ) is bounded for some positive constant s.
Then, for all times t ≥ 0,
! )
(
2
1X s
ds (f1 (t), f2 (t)) ≤ e
hpi + qis i − 1 t ds (f1,0 , f2,0 ).
2
i=1
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation V
In particular there exists a unique weak solution f (t) of the
Boltzmann equation.
Corollary
Let f1 (t) and f2 (t) be two solutions of the Boltzmann equation,
corresponding to initial values with the same mean. Assume that
ds (f1,0 , f2,0 ) is bounded for some positive constant s.
In case
!
2
1X s
hpi + qis i − 1 < 0
2
i=1
the distance ds of f1 and f2 decays exponentially in time.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The Boltzmann equation V
In particular there exists a unique weak solution f (t) of the
Boltzmann equation.
Corollary
Let f1 (t) and f2 (t) be two solutions of the Boltzmann equation,
corresponding to initial values with the same mean. Assume that
ds (f1,0 , f2,0 ) is bounded for some positive constant s.
In case
!
2
1X s
hpi + qis i − 1 < 0
2
i=1
the distance ds of f1 and f2 decays exponentially in time.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity
The large-time behavior of the solution to the kinetic equation
depends of the quantity
2
S(s) =
1X s
hpi + qis i − 1.
2
i=1
The sign of this quantity, in fact, is related both to the number
of moments of the solution which remain uniformly bounded in
time, and to the possibility to conclude the existence and
uniqueness of a steady state.
Let s > 1 be such that
Z
Ms =
v s f0 (v ) dv < ∞,
IR +
and assume further that S(s) is finite, but is not zero.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity
The large-time behavior of the solution to the kinetic equation
depends of the quantity
2
S(s) =
1X s
hpi + qis i − 1.
2
i=1
The sign of this quantity, in fact, is related both to the number
of moments of the solution which remain uniformly bounded in
time, and to the possibility to conclude the existence and
uniqueness of a steady state.
Let s > 1 be such that
Z
Ms =
v s f0 (v ) dv < ∞,
IR +
and assume further that S(s) is finite, but is not zero.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity
The large-time behavior of the solution to the kinetic equation
depends of the quantity
2
S(s) =
1X s
hpi + qis i − 1.
2
i=1
The sign of this quantity, in fact, is related both to the number
of moments of the solution which remain uniformly bounded in
time, and to the possibility to conclude the existence and
uniqueness of a steady state.
Let s > 1 be such that
Z
Ms =
v s f0 (v ) dv < ∞,
IR +
and assume further that S(s) is finite, but is not zero.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity II
Define
1
K(s) := hp1s−1 q1 + p2s−1 q2 + p1 q1s−1 + p2 q2s−1 i
2
1
≤ hp1s + p2s + q1s + q2s i
2
≤ (S(s) + 1) .
If S(s) is a finite number, then so is K(s).
The main result on S(s) gives its relationship with moments.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity II
Define
1
K(s) := hp1s−1 q1 + p2s−1 q2 + p1 q1s−1 + p2 q2s−1 i
2
1
≤ hp1s + p2s + q1s + q2s i
2
≤ (S(s) + 1) .
If S(s) is a finite number, then so is K(s).
The main result on S(s) gives its relationship with moments.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A noticeable quantity II
Define
1
K(s) := hp1s−1 q1 + p2s−1 q2 + p1 q1s−1 + p2 q2s−1 i
2
1
≤ hp1s + p2s + q1s + q2s i
2
≤ (S(s) + 1) .
If S(s) is a finite number, then so is K(s).
The main result on S(s) gives its relationship with moments.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Main result on moments
Theorem
The weak solution to the Boltzmann equation satisfies the following
estimates:
1
If S(s) > 0, then, as t → ∞,
R
s
s
θs K(s)
Θs K(s)
R+ v f (v , t) dv
1/s
s
·M ≤
·M
Ms +
≤ Ms +
S(s)
e{t · S(s)}
S(s)
2
If S(s) < 0, then the s-th moment is bounded for all times.
Moreover, as t → ∞,
Z
θs K(s)
Θs K(s) s
s
s
·M ≤
v f (v , t) dv ≤
· Ms.
|S(s)|
|S(s)|
R+
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Properties of steady states
Theorem
Suppose there exists s such that S(s) < 0, and let f∞ (v ) be the
unique stationary solution to the Boltzmann equation. Let f (v , t)
be the weak solution. Then, if the initial density f0 satisfies
Z
v σ f0 (v ) dv < ∞,
IR +
where σ = max{1, s}, f (v , t) converges exponentially fast in
Fourier metric towards f∞ (v ), and the following bound holds
ds (f (t), f∞ ) ≤ ds (f0 , f∞ )e {−|S(s)|t} .
Giuseppe Toscani
Kinetic models of economy
(4)
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Consequences
Consider a conservative trade
v ∗ = p1 v + q1 w ;
w ∗ = p2 v + q2 w .
Introduce the quantity
2
1X s
S(s) =
hpi + qis i − 1.
2
i=1
The large-time behavior of the solution to the kinetic equation
depends of the knowledge of the sign of the function S(s).
Both the existence and uniqueness of a solution and the
existence of a steady distribution depend on the microscopic
details of the binary interaction.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Consequences
Consider a conservative trade
v ∗ = p1 v + q1 w ;
w ∗ = p2 v + q2 w .
Introduce the quantity
2
1X s
S(s) =
hpi + qis i − 1.
2
i=1
The large-time behavior of the solution to the kinetic equation
depends of the knowledge of the sign of the function S(s).
Both the existence and uniqueness of a solution and the
existence of a steady distribution depend on the microscopic
details of the binary interaction.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Consequences
Consider a conservative trade
v ∗ = p1 v + q1 w ;
w ∗ = p2 v + q2 w .
Introduce the quantity
2
1X s
S(s) =
hpi + qis i − 1.
2
i=1
The large-time behavior of the solution to the kinetic equation
depends of the knowledge of the sign of the function S(s).
Both the existence and uniqueness of a solution and the
existence of a steady distribution depend on the microscopic
details of the binary interaction.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Consequences
Consider a conservative trade
v ∗ = p1 v + q1 w ;
w ∗ = p2 v + q2 w .
Introduce the quantity
2
1X s
S(s) =
hpi + qis i − 1.
2
i=1
The large-time behavior of the solution to the kinetic equation
depends of the knowledge of the sign of the function S(s).
Both the existence and uniqueness of a solution and the
existence of a steady distribution depend on the microscopic
details of the binary interaction.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s)
Differentiating twice with respect to s
2
1X s 2
d 2 S(s)
=
hpi ln pi + qis ln2 qi i > 0.
2
ds
2
i=1
S(s) is convex on IR + . S(0) = 1, and S(1) = 0
Only three possible scenarios for the qualitative behavior of S.
Characterized by the sign of S0 (1).
S0 (1) = 0 by convexity implies S(s) ≥ 0 for all s > 0. In this
case, it does not exist any s̄ ∈ IR + such that S(s̄) < 0, and
all the conclusions fail.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s)
Differentiating twice with respect to s
2
1X s 2
d 2 S(s)
=
hpi ln pi + qis ln2 qi i > 0.
2
ds
2
i=1
S(s) is convex on IR + . S(0) = 1, and S(1) = 0
Only three possible scenarios for the qualitative behavior of S.
Characterized by the sign of S0 (1).
S0 (1) = 0 by convexity implies S(s) ≥ 0 for all s > 0. In this
case, it does not exist any s̄ ∈ IR + such that S(s̄) < 0, and
all the conclusions fail.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s)
Differentiating twice with respect to s
2
1X s 2
d 2 S(s)
=
hpi ln pi + qis ln2 qi i > 0.
2
ds
2
i=1
S(s) is convex on IR + . S(0) = 1, and S(1) = 0
Only three possible scenarios for the qualitative behavior of S.
Characterized by the sign of S0 (1).
S0 (1) = 0 by convexity implies S(s) ≥ 0 for all s > 0. In this
case, it does not exist any s̄ ∈ IR + such that S(s̄) < 0, and
all the conclusions fail.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s)
Differentiating twice with respect to s
2
1X s 2
d 2 S(s)
=
hpi ln pi + qis ln2 qi i > 0.
2
ds
2
i=1
S(s) is convex on IR + . S(0) = 1, and S(1) = 0
Only three possible scenarios for the qualitative behavior of S.
Characterized by the sign of S0 (1).
S0 (1) = 0 by convexity implies S(s) ≥ 0 for all s > 0. In this
case, it does not exist any s̄ ∈ IR + such that S(s̄) < 0, and
all the conclusions fail.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) II
Suppose now S0 (1) > 0.
In this case, since S(0) = 1, while S(1) = 0, the convex
function S(s) has a minimum in the interval 0 < s < 1.
If the minimum is attained at the point s̄, S(s) < 0 at least
in the interval s̄ < s < 1, and at the same time S(s) > 0 for
s > 1.
Theorems imply that the kinetic equation has a unique steady
state that possesses moments bounded up to the order one,
while all the moments of order bigger than one are infinite.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) II
Suppose now S0 (1) > 0.
In this case, since S(0) = 1, while S(1) = 0, the convex
function S(s) has a minimum in the interval 0 < s < 1.
If the minimum is attained at the point s̄, S(s) < 0 at least
in the interval s̄ < s < 1, and at the same time S(s) > 0 for
s > 1.
Theorems imply that the kinetic equation has a unique steady
state that possesses moments bounded up to the order one,
while all the moments of order bigger than one are infinite.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) II
Suppose now S0 (1) > 0.
In this case, since S(0) = 1, while S(1) = 0, the convex
function S(s) has a minimum in the interval 0 < s < 1.
If the minimum is attained at the point s̄, S(s) < 0 at least
in the interval s̄ < s < 1, and at the same time S(s) > 0 for
s > 1.
Theorems imply that the kinetic equation has a unique steady
state that possesses moments bounded up to the order one,
while all the moments of order bigger than one are infinite.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) II
Suppose now S0 (1) > 0.
In this case, since S(0) = 1, while S(1) = 0, the convex
function S(s) has a minimum in the interval 0 < s < 1.
If the minimum is attained at the point s̄, S(s) < 0 at least
in the interval s̄ < s < 1, and at the same time S(s) > 0 for
s > 1.
Theorems imply that the kinetic equation has a unique steady
state that possesses moments bounded up to the order one,
while all the moments of order bigger than one are infinite.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) III
Suppose S0 (1) < 0.
The presence of tails in the steady distribution depend whether
lim S(s)< (>)0,
s→∞
In the < case, S(s) < 0 for s > 1. The kinetic equation has a
unique steady state that possesses all moments bounded.
In the > case, since S(1) = 0, the convex function S(s) has
a minimum attained at some point s̃ > 1, and at the same
time there exists s̄ > s̃ in which S(s̄) = 0.
The kinetic equation (4) has a unique steady state that
possesses moments bounded only up to the order s̄ (excluded)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) III
Suppose S0 (1) < 0.
The presence of tails in the steady distribution depend whether
lim S(s)< (>)0,
s→∞
In the < case, S(s) < 0 for s > 1. The kinetic equation has a
unique steady state that possesses all moments bounded.
In the > case, since S(1) = 0, the convex function S(s) has
a minimum attained at some point s̃ > 1, and at the same
time there exists s̄ > s̃ in which S(s̄) = 0.
The kinetic equation (4) has a unique steady state that
possesses moments bounded only up to the order s̄ (excluded)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) III
Suppose S0 (1) < 0.
The presence of tails in the steady distribution depend whether
lim S(s)< (>)0,
s→∞
In the < case, S(s) < 0 for s > 1. The kinetic equation has a
unique steady state that possesses all moments bounded.
In the > case, since S(1) = 0, the convex function S(s) has
a minimum attained at some point s̃ > 1, and at the same
time there exists s̄ > s̃ in which S(s̄) = 0.
The kinetic equation (4) has a unique steady state that
possesses moments bounded only up to the order s̄ (excluded)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) III
Suppose S0 (1) < 0.
The presence of tails in the steady distribution depend whether
lim S(s)< (>)0,
s→∞
In the < case, S(s) < 0 for s > 1. The kinetic equation has a
unique steady state that possesses all moments bounded.
In the > case, since S(1) = 0, the convex function S(s) has
a minimum attained at some point s̃ > 1, and at the same
time there exists s̄ > s̃ in which S(s̄) = 0.
The kinetic equation (4) has a unique steady state that
possesses moments bounded only up to the order s̄ (excluded)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
The sign of S(s) III
Suppose S0 (1) < 0.
The presence of tails in the steady distribution depend whether
lim S(s)< (>)0,
s→∞
In the < case, S(s) < 0 for s > 1. The kinetic equation has a
unique steady state that possesses all moments bounded.
In the > case, since S(1) = 0, the convex function S(s) has
a minimum attained at some point s̃ > 1, and at the same
time there exists s̄ > s̃ in which S(s̄) = 0.
The kinetic equation (4) has a unique steady state that
possesses moments bounded only up to the order s̄ (excluded)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Pointwise conservative trades
Consider models where
0 ≤ p1 , p2 , q1 , q2 ≤ 1
with probability one. hpi i =
6 0 hqi i =
6 0 for i = 1 or i = 2..
Each pointwise conservative model fits into this class.
S(s) is negative for all s > 1, which guarantees boundedness
of all moments in time.
Among others, Chakraborty - Chakrabarthi model belongs to
this class.
The solution to Chakraborty - Chakrabarthi model does not
develop tails.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Pointwise conservative trades
Consider models where
0 ≤ p1 , p2 , q1 , q2 ≤ 1
with probability one. hpi i =
6 0 hqi i =
6 0 for i = 1 or i = 2..
Each pointwise conservative model fits into this class.
S(s) is negative for all s > 1, which guarantees boundedness
of all moments in time.
Among others, Chakraborty - Chakrabarthi model belongs to
this class.
The solution to Chakraborty - Chakrabarthi model does not
develop tails.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Pointwise conservative trades
Consider models where
0 ≤ p1 , p2 , q1 , q2 ≤ 1
with probability one. hpi i =
6 0 hqi i =
6 0 for i = 1 or i = 2..
Each pointwise conservative model fits into this class.
S(s) is negative for all s > 1, which guarantees boundedness
of all moments in time.
Among others, Chakraborty - Chakrabarthi model belongs to
this class.
The solution to Chakraborty - Chakrabarthi model does not
develop tails.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Pointwise conservative trades
Consider models where
0 ≤ p1 , p2 , q1 , q2 ≤ 1
with probability one. hpi i =
6 0 hqi i =
6 0 for i = 1 or i = 2..
Each pointwise conservative model fits into this class.
S(s) is negative for all s > 1, which guarantees boundedness
of all moments in time.
Among others, Chakraborty - Chakrabarthi model belongs to
this class.
The solution to Chakraborty - Chakrabarthi model does not
develop tails.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Pointwise conservative trades
Consider models where
0 ≤ p1 , p2 , q1 , q2 ≤ 1
with probability one. hpi i =
6 0 hqi i =
6 0 for i = 1 or i = 2..
Each pointwise conservative model fits into this class.
S(s) is negative for all s > 1, which guarantees boundedness
of all moments in time.
Among others, Chakraborty - Chakrabarthi model belongs to
this class.
The solution to Chakraborty - Chakrabarthi model does not
develop tails.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades
Consider models which are conservative in the mean only .
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is of this
type
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
Assume that the probability variables η and η̃ are independent
and identically distributed
Under these assumptions,
S(s) = hp1s + p2s i − 1 = (1 − λ)s − 1 +
Z
+∞
(λ + ω)s dρ(s),
−λ
where ρ(s) is the probability density of η and η̃ on [−λ, +∞).
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades
Consider models which are conservative in the mean only .
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is of this
type
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
Assume that the probability variables η and η̃ are independent
and identically distributed
Under these assumptions,
S(s) = hp1s + p2s i − 1 = (1 − λ)s − 1 +
Z
+∞
(λ + ω)s dρ(s),
−λ
where ρ(s) is the probability density of η and η̃ on [−λ, +∞).
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades
Consider models which are conservative in the mean only .
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is of this
type
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
Assume that the probability variables η and η̃ are independent
and identically distributed
Under these assumptions,
S(s) = hp1s + p2s i − 1 = (1 − λ)s − 1 +
Z
+∞
(λ + ω)s dρ(s),
−λ
where ρ(s) is the probability density of η and η̃ on [−λ, +∞).
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades
Consider models which are conservative in the mean only .
In [S. Cordier, L. Pareschi, G. Toscani (2005)] the trade is of this
type
v ∗ = λv + (1 − λ)w + ηv
w ∗ = (1 − λ)v + λw + η̃w .
Assume that the probability variables η and η̃ are independent
and identically distributed
Under these assumptions,
S(s) = hp1s + p2s i − 1 = (1 − λ)s − 1 +
Z
+∞
(λ + ω)s dρ(s),
−λ
where ρ(s) is the probability density of η and η̃ on [−λ, +∞).
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades II
Assume η only takes values ±µ, with probability
where 0 ≤ µ ≤ λ.
1
2
each,
By varying the parameters λ and µ, one encounters the whole
variety of possible behaviors of the moments of f∞ .
If λ + µ ≤ 1, then the trade is pointwise conservative and all
moments are finite.
Assume λ + µ > 1. Evaluate
1
1
S(s) = (1 − λ)s − 1 + (λ − µ)s + (λ + µ)s .
2
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades II
Assume η only takes values ±µ, with probability
where 0 ≤ µ ≤ λ.
1
2
each,
By varying the parameters λ and µ, one encounters the whole
variety of possible behaviors of the moments of f∞ .
If λ + µ ≤ 1, then the trade is pointwise conservative and all
moments are finite.
Assume λ + µ > 1. Evaluate
1
1
S(s) = (1 − λ)s − 1 + (λ − µ)s + (λ + µ)s .
2
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades II
Assume η only takes values ±µ, with probability
where 0 ≤ µ ≤ λ.
1
2
each,
By varying the parameters λ and µ, one encounters the whole
variety of possible behaviors of the moments of f∞ .
If λ + µ ≤ 1, then the trade is pointwise conservative and all
moments are finite.
Assume λ + µ > 1. Evaluate
1
1
S(s) = (1 − λ)s − 1 + (λ − µ)s + (λ + µ)s .
2
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Conservative in the mean trades II
Assume η only takes values ±µ, with probability
where 0 ≤ µ ≤ λ.
1
2
each,
By varying the parameters λ and µ, one encounters the whole
variety of possible behaviors of the moments of f∞ .
If λ + µ ≤ 1, then the trade is pointwise conservative and all
moments are finite.
Assume λ + µ > 1. Evaluate
1
1
S(s) = (1 − λ)s − 1 + (λ − µ)s + (λ + µ)s .
2
2
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
Tails in CPT model
In region II, all moments remain bounded for all times, while
in regions III and IV, sufficiently high or all moments above
one, respectively, diverge. Notice that the parameter region I
is excluded by the condition µ ≤ λ.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution
Easy to identify models for which all moments remain
bounded for all times. In pointwise conservative trades the
appearance of Pareto tails is excluded.
Hard to decide from the output of numerical experiments on
the nature of the tail – exponential or Pareto.
If S is negative but very close to zero, the lower bounds for
the moments become huge.
The CC-model with constant exchange rate =
very low or very high saving rates λ.
S(s) =
1+λ
2
s
Giuseppe Toscani
+
1−λ
2
s
− 1.
Kinetic models of economy
1
2
and either
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution
Easy to identify models for which all moments remain
bounded for all times. In pointwise conservative trades the
appearance of Pareto tails is excluded.
Hard to decide from the output of numerical experiments on
the nature of the tail – exponential or Pareto.
If S is negative but very close to zero, the lower bounds for
the moments become huge.
The CC-model with constant exchange rate =
very low or very high saving rates λ.
S(s) =
1+λ
2
s
Giuseppe Toscani
+
1−λ
2
s
− 1.
Kinetic models of economy
1
2
and either
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution
Easy to identify models for which all moments remain
bounded for all times. In pointwise conservative trades the
appearance of Pareto tails is excluded.
Hard to decide from the output of numerical experiments on
the nature of the tail – exponential or Pareto.
If S is negative but very close to zero, the lower bounds for
the moments become huge.
The CC-model with constant exchange rate =
very low or very high saving rates λ.
S(s) =
1+λ
2
s
Giuseppe Toscani
+
1−λ
2
s
− 1.
Kinetic models of economy
1
2
and either
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution
Easy to identify models for which all moments remain
bounded for all times. In pointwise conservative trades the
appearance of Pareto tails is excluded.
Hard to decide from the output of numerical experiments on
the nature of the tail – exponential or Pareto.
If S is negative but very close to zero, the lower bounds for
the moments become huge.
The CC-model with constant exchange rate =
very low or very high saving rates λ.
S(s) =
1+λ
2
s
Giuseppe Toscani
+
1−λ
2
s
− 1.
Kinetic models of economy
1
2
and either
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution II
For each fixed s > 2, this expression converges to zero for
either λ → 0 (no saving) or λ → 1 (no trading).
The time-uniform lower bound on the sth moment blows up
like the inverse of S(s).
Numerically one could mistake the dramatic temporal growth
of moments for s > 2 (and most probably also for all s > 1)
as convergence to a steady state f∞ with unbounded
moments corresponding to a Pareto tail.
Since the value of the first moment is fixed by the total
wealth, but all the higher moments seemingly diverge, one
would most likely “observe” a fat tail distribution with critical
Pareto index ν = 1.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution II
For each fixed s > 2, this expression converges to zero for
either λ → 0 (no saving) or λ → 1 (no trading).
The time-uniform lower bound on the sth moment blows up
like the inverse of S(s).
Numerically one could mistake the dramatic temporal growth
of moments for s > 2 (and most probably also for all s > 1)
as convergence to a steady state f∞ with unbounded
moments corresponding to a Pareto tail.
Since the value of the first moment is fixed by the total
wealth, but all the higher moments seemingly diverge, one
would most likely “observe” a fat tail distribution with critical
Pareto index ν = 1.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution II
For each fixed s > 2, this expression converges to zero for
either λ → 0 (no saving) or λ → 1 (no trading).
The time-uniform lower bound on the sth moment blows up
like the inverse of S(s).
Numerically one could mistake the dramatic temporal growth
of moments for s > 2 (and most probably also for all s > 1)
as convergence to a steady state f∞ with unbounded
moments corresponding to a Pareto tail.
Since the value of the first moment is fixed by the total
wealth, but all the higher moments seemingly diverge, one
would most likely “observe” a fat tail distribution with critical
Pareto index ν = 1.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
The Boltzmann equation
Formation of tails
Examples
A note on caution II
For each fixed s > 2, this expression converges to zero for
either λ → 0 (no saving) or λ → 1 (no trading).
The time-uniform lower bound on the sth moment blows up
like the inverse of S(s).
Numerically one could mistake the dramatic temporal growth
of moments for s > 2 (and most probably also for all s > 1)
as convergence to a steady state f∞ with unbounded
moments corresponding to a Pareto tail.
Since the value of the first moment is fixed by the total
wealth, but all the higher moments seemingly diverge, one
would most likely “observe” a fat tail distribution with critical
Pareto index ν = 1.
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
We discussed the large–time behavior of various kinetic
models of conservative economies.
It has been shown that the tails of the (unique) stationary
solution depend on the details of the microscopic interaction.
The main difference in the large-time evolution of the solution
relies is related to the fact that the trade is or not pointwise
conservative.
Pointwise conservative trades produce a steady state which
possesses all moments bounded.
Pareto tails can appear only on conservative in the mean
trades
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
We discussed the large–time behavior of various kinetic
models of conservative economies.
It has been shown that the tails of the (unique) stationary
solution depend on the details of the microscopic interaction.
The main difference in the large-time evolution of the solution
relies is related to the fact that the trade is or not pointwise
conservative.
Pointwise conservative trades produce a steady state which
possesses all moments bounded.
Pareto tails can appear only on conservative in the mean
trades
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
We discussed the large–time behavior of various kinetic
models of conservative economies.
It has been shown that the tails of the (unique) stationary
solution depend on the details of the microscopic interaction.
The main difference in the large-time evolution of the solution
relies is related to the fact that the trade is or not pointwise
conservative.
Pointwise conservative trades produce a steady state which
possesses all moments bounded.
Pareto tails can appear only on conservative in the mean
trades
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
We discussed the large–time behavior of various kinetic
models of conservative economies.
It has been shown that the tails of the (unique) stationary
solution depend on the details of the microscopic interaction.
The main difference in the large-time evolution of the solution
relies is related to the fact that the trade is or not pointwise
conservative.
Pointwise conservative trades produce a steady state which
possesses all moments bounded.
Pareto tails can appear only on conservative in the mean
trades
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
We discussed the large–time behavior of various kinetic
models of conservative economies.
It has been shown that the tails of the (unique) stationary
solution depend on the details of the microscopic interaction.
The main difference in the large-time evolution of the solution
relies is related to the fact that the trade is or not pointwise
conservative.
Pointwise conservative trades produce a steady state which
possesses all moments bounded.
Pareto tails can appear only on conservative in the mean
trades
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
Our analysis has been largely based on the possibility to apply
Fourier transform techniques.
This analysis is reminiscent of previous results for the Boltzmann
equation for Maxwell molecules
L. Pareschi, G. Toscani, Self-similarity and power-like tails in
nonconservative kinetic models, J. Statist. Phys.124 747-779 (2006)
D. Matthes, G. Toscani, On steady distributions of kinetic models of
conservative economies, (preprint) (2006)
http://www-dimat.unipv.it/toscani/
J.A. Carrillo, S. Cordier, G. Toscani, Steady distributions of
conservative-in-mean Maxwell-type models of the Boltzmann equation
(in preparation) (2007)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
Our analysis has been largely based on the possibility to apply
Fourier transform techniques.
This analysis is reminiscent of previous results for the Boltzmann
equation for Maxwell molecules
L. Pareschi, G. Toscani, Self-similarity and power-like tails in
nonconservative kinetic models, J. Statist. Phys.124 747-779 (2006)
D. Matthes, G. Toscani, On steady distributions of kinetic models of
conservative economies, (preprint) (2006)
http://www-dimat.unipv.it/toscani/
J.A. Carrillo, S. Cordier, G. Toscani, Steady distributions of
conservative-in-mean Maxwell-type models of the Boltzmann equation
(in preparation) (2007)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
Our analysis has been largely based on the possibility to apply
Fourier transform techniques.
This analysis is reminiscent of previous results for the Boltzmann
equation for Maxwell molecules
L. Pareschi, G. Toscani, Self-similarity and power-like tails in
nonconservative kinetic models, J. Statist. Phys.124 747-779 (2006)
D. Matthes, G. Toscani, On steady distributions of kinetic models of
conservative economies, (preprint) (2006)
http://www-dimat.unipv.it/toscani/
J.A. Carrillo, S. Cordier, G. Toscani, Steady distributions of
conservative-in-mean Maxwell-type models of the Boltzmann equation
(in preparation) (2007)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
Our analysis has been largely based on the possibility to apply
Fourier transform techniques.
This analysis is reminiscent of previous results for the Boltzmann
equation for Maxwell molecules
L. Pareschi, G. Toscani, Self-similarity and power-like tails in
nonconservative kinetic models, J. Statist. Phys.124 747-779 (2006)
D. Matthes, G. Toscani, On steady distributions of kinetic models of
conservative economies, (preprint) (2006)
http://www-dimat.unipv.it/toscani/
J.A. Carrillo, S. Cordier, G. Toscani, Steady distributions of
conservative-in-mean Maxwell-type models of the Boltzmann equation
(in preparation) (2007)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
Conclusions
Our analysis has been largely based on the possibility to apply
Fourier transform techniques.
This analysis is reminiscent of previous results for the Boltzmann
equation for Maxwell molecules
L. Pareschi, G. Toscani, Self-similarity and power-like tails in
nonconservative kinetic models, J. Statist. Phys.124 747-779 (2006)
D. Matthes, G. Toscani, On steady distributions of kinetic models of
conservative economies, (preprint) (2006)
http://www-dimat.unipv.it/toscani/
J.A. Carrillo, S. Cordier, G. Toscani, Steady distributions of
conservative-in-mean Maxwell-type models of the Boltzmann equation
(in preparation) (2007)
Giuseppe Toscani
Kinetic models of economy
Outlines
Formation of Pareto tails
Kinetic models
Conclusions
References
Baldassarri A., Marini Bettolo Marconi U., Puglisi A. :
Kinetic models of inelastic gases.
Mat. Mod. Meth. Appl. Sci. 12 965–983 (2002).
Ben-Avraham D., Ben-Naim E., Lindenberg K., Rosas A.:
Self-similarity in random collision processes,
Phys. Rev. E, 68, R050103 (2003).
Ben-Naim E., Krapivski P. :
Multiscaling in inelastic collisions.
Phys. Rev. E, 61, R5–R8 (2000).
A.V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell
molecules Sov. Sci. Review C 7, 112-229 (1988).
Bobylev A.V., Cercignani C.:
Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions.
J. Statist. Phys. 110, 333-375 (2003).
Bobylev, A. V.; Gamba, I. M.; Panferov, V. A.
Moment inequalities and high-energy tails for Boltzmann equations with inelastic
interactions. J. Statist. Phys. 116 (2004)
Bouchaud J.P., Mézard M.:
Wealth condensation in a simple model of economy, Physica A, 282, 536- (2000).
J.A. Carrillo, J.L. Vázquez,
Fine asymptotics for fast diffusion equations.
Comm. Partial Differential Equations 28 (2003), 1023–1056.
A. Chakraborti, B.K. Chakrabarti, Statistical mechanics of money: how saving propensity
affects its distributions, Eur. Phys. J. B., 17, 167-170 (2000).
A. Chatterjee, B.K. Chakrabarti, S.S. Manna, Pareto law in a kinetic model of market with
random saving propensity. Physica A 335 155-163 (2004)
S. Cordier, L. Pareschi, G. Toscani,
On a kinetic model for a simple market economy.
J. Stat. Phys. 120 253-277 (2005).
T. Di Matteo, T. Aste, S.T. Hyde, in The physics of complex systems (New advances and
perspectives) F, Mallamace & H.E. Stanley eds., IOS press, Amsterdam, 2004
A.Dragulescu, V.M.Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17,
723–729 (2000).
A.Dragulescu, V.M.Yakovenko, Exponential and power law probability distribution of wealth
and income in the United kingdom and the United states , Physica A, 299, 213-221 (2001).
H.P. McKean,
The central limit theorem for Carleman’s equation,
Israel J. Math., 21 (1975) 54-92.
B.Mandelbrot, The Pareto-Lévy law and the distribution of income, International Economic
Review ,1, 79-106 (1960).
V.Pareto, Cours d’Economie Politique, Lausanne and Paris, (1897).
F. Slanina, Inelastically scattering particles and wealth distribution in an open economy,
Phys. Rev. E 69, 046102 (2004).
Giuseppe Toscani
Kinetic models of economy