Transcript Power Law Tails in the Italian Personal Income Distribution

Power Law Tails in the Italian
Personal Income Distribution
F. Clementi1,3 and M. Gallegati2,3
1Department
of Public Economics, University of Rome “La Sapienza”, Via del
Castro Laurenziano 9, I–00161 Rome, Italy
[email protected]
2Department
3S.I.E.C.,
of Economics, Università Politecnica delle Marche, Piazzale
Martelli
8, I–62100 Ancona, Italy
[email protected]
Università Politecnica delle Marche, Piazzale Martelli 8, I–62100
Ancona, Italy
http://www.dea.unian.it/wehia/
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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1. Introduction
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
PARETO LAW. More than a century ago the Italian economist Vilfredo Pareto stated in his
Cours d'Économie Politique (1897) that a plot of the logarithm of the number of incomereceiving units above a certain threshold against the logarithm of the income yields points
close to a straight line.

RECENT EMPIRICAL WORK. Recent empirical work seems to confirm the validity of
Pareto (power) law. For example, Aoyama et al. (2000) show that the distribution of income
and income tax of individuals in Japan for the year 1998 is very well fitted by a Pareto powerlaw type distribution, even if it gradually deviates as the income approaches lower ranges.
The applicability of Pareto distribution only to high incomes is actually acknowledged;
therefore, other kinds of distributions has been proposed by researchers for the low-middle
income region. According to Montroll and Shlesinger (1983), US personal income data for the
years 1935-36 suggest a power-law distribution for the high-income range and a lognormal
distribution for the rest; a similar shape is found by Souma (2001) investigating the Japanese
income and income tax data for the high-income range over the 112 years 1887-1998, and for
the middle-income range over the 44 years 1955-98. Nirei and Souma (2004) confirm the
power-law decay for top taxpayers in the US and Japan from 1960 to 1999, but find that the
middle portion of the income distribution has rather an exponential form; the same is
proposed by Drăgulescu and Yakovenko (2001) for the UK during the period 1994-99 and for
the US in 1998.

THE AIM OF THIS ANALYSIS. We look at the shape of the personal income distribution in
Italy by using cross-sectional data samples from the population of Italian households during
the years 1977-2002. We find that the personal income distribution follows the Pareto law in
the high-income range, while the lognormal pattern is more appropriate in the central body of
the distribution. From this analysis we get the result that the indexes specifying the
distribution change in time; therefore, we try to look for some factors which might be the
potential reasons for this behaviour.
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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2. Lognormal Pattern
with Power Law Tail
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2.1 The Data Source

DATA SOURCE. The Historical Archive (HA) of the Survey on Household Income and
Wealth (SHIW), made publicly available by the Bank of Italy for the period 1977-2002, was
carried out yearly until 1987 (except for 1985) and every two years thereafter (the survey for
1997 was shifted to 1998).

DEFINITION OF INCOME. The basic definition of income provided by the SHIW is net of
taxation and social security contributions. It is the sum of four main components:
compensation of employees; pensions and net transfers; net income from self-employment;
property income (including income from buildings and income from financial assets). Income
from financial assets started to be recorded only in 1987.

SAMPLE SIZE. The average number of income-earners surveyed from the SHIW-HA is
about 10,000.

CURRENCY UNIT. All amounts are expressed in thousands of lire, except for 2002, where
incomes are reported in euros.
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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2.2 Empirical Findings
 1  ln  x     2 
f  x ,  
exp  
 

x 2
 
 2 
1
0
-2
Income data
Lognormal
Power law
-4
LOGNORMAL PATTERN... The profile
of the personal income distribution for the
year 1998 suggests that the central body of
the distribution (almost all of it below the
99th percentile) follows a two-parameter
lognormal distribution:
Cumulative probability

2
The cumulative probability distribution of the Italian personal income in 1998
1
2
3
Income (thousand £)
4
5
f  x ,k  
 k
x 1
0
-1
-2
…WITH POWER-LAW TAIL. On the
contrary, the tail of the distribution
(including about the top 1% of the
population) follows a Pareto (power-law)
distribution:
Cumulative probability

1
The fit to the power-law distribution for the year 1998
Income data
Power law
3.5
4
4.5
Income (thousand £)
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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3. Time Development
of the Distribution
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3.1 Temporal Change of the Distribution
1
0
1977/78/79
1980/81/82/83/84/86/87/89
1991/93/95/98
2000/02
-2

-1
UNIVERSAL STRUCTURE. The
distribution pattern of the personal income
expressed as the lognormal with powerlaw tail seems to hold all over the years
covered by our data set.
Cumulative probability

2
Time development of the Italian personal income distribution over the
years 1977-2002
ESTIMATION RESULTS. The
estimation results show a shift of the
distribution and a change of the indexes
specifying it. This fact means that the
curvature of the lognormal fit and the
power-law slope differ from year to year,
i.e. Gibrat index (measured as β=1/(σ√2))
and Pareto index change in time.
2.4
3
4
Income (thousand £)
5
Estimated lognormal and Pareto distribution parameters for all the
years
Year
1977
1978
1979
1980
1981
1982
1983
1984
1986
1987
1989
1991
1993
1995
1998
2000
2002
ˆ
3.31
3.33
3.34
3.36
3.36
3.38
3.38
3.39
3.40
3.49
3.53
3.52
3.47
3.46
3.48
3.50
3.52
(0.005)
(0.005)
(0.005)
(0.005)
(0.005)
(0.004)
(0.004)
(0.004)
(0.004)
(0.004)
(0.003)
(0.004)
(0.004)
(0.004)
(0.004)
(0.004)
(0.004)
ˆ
0.34
0.34
0.34
0.33
0.32
0.31
0.30
0.32
0.29
0.30
0.26
0.27
0.33
0.32
0.34
0.32
0.31
(0.004)
(0.004)
(0.005)
(0.005)
(0.004)
(0.005)
(0.004)
(0.005)
(0.006)
(0.004)
(0.003)
(0.004)
(0.004)
(0.003)
(0.006)
(0.004)
(0.005)
ˆ
2.08
2.09
2.08
2.15
2.23
2.27
2.32
2.24
2.40
2.38
2.70
2.58
2.15
2.19
2.10
2.20
2.25
ˆ
3.00
3.01
2.91
3.06
3.30
3.08
3.11
3.05
3.04
2.09
2.91
3.45
2.74
2.72
2.76
2.76
2.71
(0.008)
(0.008)
(0.009)
(0.008)
(0.008)
(0.005)
(0.006)
(0.007)
(0.005)
(0.002)
(0.002)
(0.008)
(0.002)
(0.002)
(0.002)
(0.002)
(0.002)
ˆ0
x
R2
10,876
11,217
11,740
11,453
10,284
11,456
11,147
11,596
11,597
24,120
15,788
14,281
16,625
16,587
17,141
17,470
17,664
0.9921
0.9933
0.9908
0.9915
0.9939
0.9952
0.9945
0.9937
0.9950
0.9993
0.9995
0.9988
0.9997
0.9996
0.9993
0.9994
0.9997
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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3.2 The Shift of the Distribution: GDP and
Personal Income Growth Rate Distributions
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Probability density function of Italian annual GDP growth rates for the
period 1977-2002
.6
.8
GDP data (1977-2002)
Laplace fit
-.04
-.02
0
Growth rate
 x 
f  x  ,  
exp  

 
 2

RPI  log  PIt i PIt 
.8
.6
.4
.2
0
Probability density
...AND PERSONAL INCOME (PI)
GROWTH RATE DISTRIBUTION. the
same functional form seems to be valid
also in the case of PI growth rates:
.04
PI data (1989/1987)
Laplace fit
1

.02
Probability distribution of Italian PI growth rate between 1989 and 1987
1
By means of a non-linear algorithm, we
find that the probability density function
of annual GDP growth rates is well fitted
by a Laplace distribution:
0
.2
.4
ANNUAL GDP… Macroeconomics
argues that the origin of the shift of the
distribution consists in the growth of the
Gross Domestic Product (GDP). To
confirm this hypothesis we study the
fluctuations in the growth rate of annual
GDP:
RGDP  log  GDPt 1 GDPt 
Probability density

-2
-1
0
Growth rate
1
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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3.3 The Shift of the Distribution: Universal
Features in the GDP and Personal Income
Growth Dynamics
Probability density

1
.8
 RGDP   GDP
.6
and  RPI  RPI   PI
the resulting empirical distributions appear
similar for GDP and PI growth rates. This
effect raises the intriguing possibility that a
common mechanism might characterize the
growth dynamics of both the quantities,
pointing in this way to the existence of
correlation between them.
TWO-SAMPLE KOLMOGOROVSMIRNOV TEST. To confirm this
assumption, we test the hypothesis that the
GDP and PI growth rate distributions are
the same by performing a two-sample
Kolmogorov-Smirnov test. In all the cases
we studied, the null hypothesis that the
growth rates of both quantities are sample
from the same distribution can not be
rejected at the usual 5% marginal
significance level.
GDP
GDP (1977-2002)
PI (1989/1987)
PI (1991/1989)
PI (1993/1991)
PI (1995/1993)
PI (1998/1995)
PI (2000/1998)
PI (2002/2000)
Laplace fit
.4
R
Rescaled probability distribution of Italian GDP and PI growth rates
.2
RESCALED GDP AND PI GROWTH
RATE DISTRIBUTION. After
normalization:
0

-4
-2
0
Growth rate
2
4
Estimated Kolmogorov-Smirnov test p-values for both GDP and PI
growth rate data
Growrh rate
RGDP
R89/87
R91/89
R93/91
R95/93
R98/95
R00/98
R89/87
0.872
R91/89
0.919
0.998
R93/91
0.998
0.984
0.970
R95/93
0.696
0.431
0.979
0.839
R98/95
0.337
0.689
0.995
0.459
0.172
R00/98
0.480
0.860
0.994
0.750
0.459
0.703
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
R02/00
0.955
0.840
0.997
1.000
0.560
0.378
0.658
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3.4 The Fluctuations of the Indexes
Specifying the Income Distribution
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LINK WITH THE BUSINESS CYCLE.
Although the frequency of data (initially
annual and then biennial from 1987)
makes it difficult to establish a link with
the business cycle, it seems possible to
find a (negative) relationship between the
Gibrat and Pareto indexes and the
fluctuations of economic activity, at least
until the late 1980s.
2.5
Pareto index

3.5
The temporal change of Pareto index over the years 1977-2002
2
Excluding financial assets
1977
1980
1985
1990
Year
1995
2000 2002
2.8
The temporal change of Gibrat index over the years 1977-2002
2.6
Excluding financial assets
2.2
2.4
Including financial assets
2
THE ITALIAN EXPERIENCE. For
example, Italy experienced a period of
economic growth until the late 1980s, but
with alternating phases of the internal
business cycle: of slowdown of production
up to the 1983 stagnation; of recovery in
1984; again of slowdown in 1986. The
values of Gibrat and Pareto indexes,
inferred from the numerical fitting, tend to
decrease in the periods of economic
expansion (concentration goes up) and
increase during the recessions (income is
more evenly distributed).
Gibrat index

Including financial assets
1977
1980
1985
1990
Year
1995
2000 2002
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3.5 Time Pattern of Income Inequality
GINI COEFFICIENT. The temporal change of Gini coefficient for the considered years
shows that in Italy the level of inequality decreased significantly during the 1980s and rised in
the early 1990s; it was substantially stable in the following years. In particular, a sharp rise of
Gini coefficient (i.e., of inequality) is encountered in 1987 and 1993, corresponding to a sharp
decline of Pareto index in the former case and of both Pareto and Gibrat indexes in the latter
case.
.3
.32
.34
.36
Gini coefficient for Italian personal income during the period 1977-2002
Excluding financial assets
.28

Including financial assets
1977
1980
1985
1990
Year
1995
2000 2002
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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3.6 Asset Price and Economic Performance
.5
0
-.5
-1
SPECULATIVE BUBBLE. We consider
that the decline of Pareto exponent in 1987
corresponds with the peak of the
speculative bubble begun in the early
1980s, and the rebounce of the index
follows its burst on October 19, when the
Dow Jones index lost more than 20% of its
value dragging into disaster the other
world markets. This assumption seems
confirmed by the movement of asset price
in the Italian Stock Exchange.
MIB

Temporal change of Italian Stock Exchange MIB Index during the
period 1977-2002
1977
1980
1985
1990
Year
1995
2000 2002
5.9
5.95
5.8
5.85
THE 1993 RECESSION OF
ECONOMIC ACTIVITY. As regards the
sharp decline of both indexes in 1993, the
level and growth of personal income
(especially in the middle-upper income
range) were notably influenced by the bad
results of the real economy in that year,
which induced an increase in inequality.
Gross Domestic Product

6
Temporal change of Italian GDP during the period 1977-2002
1977
1980
1985
1990
Year
1995
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
2000 2002
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3.7 Breakdown of Pareto Law
DEVIATION FROM PARETO LAW. We show that these facts (the 1987 burst of the assetinflation bubble begun in the early 1980s and the 1993 recession year) cause the invalidity of
Pareto law for high incomes; that is: during the mentioned years the data can not be fitted by a
power-law in the entire high-income range.
The fit to the power-law distribution in 1987
0
-1
-1
0
Cumulative probability
1
1
2
The fit to the power-law distribution in 1993
3.8
4
Income data
Power law
-2
Income data
Power law
-2

4.2
4.4
Income (thousand £)
4.6
4.8
3.5
4
Income (thousand £)
4.5
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4. Summary
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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
THE SHAPE OF THE INCOME DISTRIBUTION. We find that the Italian personal
income microdata are consistent with a Pareto-power law behaviour in the high-income range,
and with a two-parameter lognormal pattern in the low-middle income region.

THE SHIFT OF THE DISTRIBUTION. The numerical fitting over the time span covered
by our dataset show a shift of the distribution, which is claimed to be a consequence of the
growth of the country. This assumption is confirmed by testing the hypothesis that the growth
dynamics of both gross domestic product of the country and personal income of individuals is
the same; the two-sample Kolmogorov-Smirnov test we perform on this subject lead us to
accept the null hypothesis that the growth rates of both the quantities are samples from the
same probability distribution in all the cases we studied, pointing to the existence of
correlation between them.

TEMPORAL EVOLUTION OF GIBRAT AND PARETO INDEXES OVER THE
BUSINESS CYCLE. By calculating the yearly estimates of Pareto and Gibrat indexes, we
quantify the fluctuations of the shape of the distribution over time by establishing some links
with the business cycle phases which Italian economy experienced over the years of our
concern. We find that there exists a negative relationship between the above-stated indexes
and the fluctuations of economic activity at least until the late 1980s.

BUSINESS CYCLE EPISODES AND BREAKDOWN OF PARETO LAW. In two
circumstances (the 1987 burst of the speculative bubble begun in the early 1980s and the 1993
recession year) the data can not be fitted by a power law in the entire high-income range,
causing breakdown of Pareto law.
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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4. Forthcoming
Events
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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1)
COMPLEXITY, HETEROGENEITY AND INTERACTIONS IN ECONOMICS
AND FINANCE (CHIEF). Ancona, Italy, May 2-21, 2005:
http://www.dea.unian.it/wehia/AnconaTI_3.htm
2)
10th ANNUAL WORKSHOP ON ECONOMICS WITH HETEROGENEOUS AND
INTERACTING AGENTS (WEHIA 2005). Colchester, UK, June 13-15, 2005:
http://www.essex.ac.uk/wehia05/
3)
ECONOPOHYSICS COLLOQUIM. Canberra, Australia, November 14-18, 2005:
http://www.rsphysse.anu.edu.au/econophysics/index.php
4)
WORKSHOP ON INDUSTRY AND LABOR DYNAMICS. THE AGENT-BASED
COMPUTATIONAL ECONOMICS APPROACH (WILD@ACE). Ancona, Italy,
December 2-3, 2005: http://www.dea.unian.it/wehia/
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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Thank you all!
International Workshop on "Econophysics of Wealth Distributions", Saha Institute of Nuclear Physics, Kolkata, India, 15-19 March, 2005
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