Modelling personal incomes in the USA as a physical system

1. First meeting of the Society for the
Study of Economic Inequality (ECINEQ)
Palma de Mallorca, July 20-22, 2005
Evolution of the personal income
distribution in the USA: high incomes
Ivan O. Kitov
e-mail - ikitov@mail.ru
http://www.ecokitov.narod.ru

2. ‘The ECINEQ conference will provide a forum for a
rigorous analysis of inequality, welfare and
redistribution issues, both at the theoretical and at the
empirical level, as well as for a discussion of the policy
implications of the research findings in this field.’
Does the inequality have economic roots?

3. Evolution of the personal income distribution in the USA: high incomes
• Personal income distribution in the USA: observations and features to model
• Pareto income distribution as a realization of self-organized criticality
• The model - from microeconomic approach to macroeconomic model
• Defining the Pareto threshold
• Results from modelling: the PID in the USA is predefined
• Conclusions

4. 2004; http://www.census.gov/hhes/income/dinctabs.html)
Xm
Two branches: “subcritical” and
“supercritical”=Pareto distribution=High Incomes
Pareto threshold = Xm
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Personal income distribution in the USA: observations and features to model
Personal Income Distribution in the USA: 1994-2001
(Data Source:.U.S. Census Bureau; "Detailed Income Tabulations from the CPS". Last Revised: August 26
Evolution of the PID in the USA: high incomes
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5. Evolution of the number of people with income
above $100K: 1994-2002
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Personal income distribution in the USA: observations and features to model
A linear increase with nominal GDP growth is expected
and observed
Evolution of the PID in the USA: high incomes
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6. Personal income distribution in the USA: observations and features to model
Normalized PIDs for incomes above $100K
as a function of work experience
The distributions are very similar
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Evolution of the PID in the USA: high incomes
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7. Personal income distribution in the USA: observations and features to model
Normalized PIDs for incomes above $100K
as a function of work experience - time dependence
Variation is minor.
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Evolution of the PID in the USA: high incomes
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8. Evolution of the normalized PID in 1994 for incomes above a given
threshold: >$0K (all personal incomes), >$10K, …, >$100K
Pareto income distribution as a realization of self-organized criticality
The distributions for income above $60K are very similar:
SELF ODGANIZED CRITICALITY (SOC) - behaviour above (Pareto) threshold
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Evolution of the PID in the USA: high incomes
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9. •
•
•
•
•
(1)
M(t) is money income denominated in dollars [$]
t is the work experience expressed in some appropriate time units [time unit]
σ(t) is the capability to earn money [$/(time unit)]
α is the dissipation coefficient expressed in units [$/(time unit)]
Λ is the size of the earning means expressed in [$]
The Solution
M(t)=(σ/α)Λ(1- exp(-αt/Λ))
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(2)
The model - from microeconomic approach to macroeconomic model
The model (1)
individual income - microeconomic model
dM(t)/dt= σ(t)-αM(t)/Λ(t)
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10. The model (2)
individual income - microeconomic model
•
Σ(t)=σ(t)/α
Σ(t)= Σ(0)sqrt(GDP(t)/GDP(0))
Λ(t)= Λ(0)sqrt(GDP(t)/GDP(0)
GDP(t) is real per capita GDP
The Solution
M(t)=Σ Λ(1- exp(-αt/Λ))
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(2’)
The model - from microeconomic approach to macroeconomic model
Substitution
Evolution law
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11. The model (3)
aggregate income - macroeconomic model
Substitution
S(t)=Σ(t)/Σ min(t)
L(t)=Λ(t)/Λ min(t)
Distribution
S1=2,…, S29=30
L1=2,…, L29=30
The model - from microeconomic approach to macroeconomic model
Σ min(t) and Λ min(t) are the nonzero minimum values for all the persons
of the modified effective capability to earn money,
Σ(t)
and the size of earning means, Λ(t)
SiLj={2×2/900, 2×3/900, …, 2×30/900, 3×2/900, …, 3×30/900, …,
...
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..., 30×30/900}.
Evolution of the PID in the USA: high incomes
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12. The model (4)
aggregate income - macroeconomic model
(1-exp(-(α/Λ minLmax)t/(Lk/Lmax))) (3)
Normalized form
M'ik(t) =Σ min(t)Λ min(t)S'iL'j{1-exp[-(1/Λ min)(α't/L‘k)]}
(3’)
Σ min (t)= Σ min (0)sqrt(GDP(t)/GDP(0))
The model - from microeconomic approach to macroeconomic model
Mik(t)/(SmaxLmax)=(Σ min Λ min)(Si/Smax)(Lk/Lmax)
Λ min (t)= Λ min (0)sqrt(GDP(t)/GDP(0)
Σ min (0)= Λ min (0)=1
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13. The model (5)
The model - from microeconomic approach to macroeconomic model
Only the persons with high L and S can reach the Pareto threshold
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14. Mean personal income vs. work experience
(Data Source: The U.S. Census Bureau, www.census.gov )
Tcr
Τ cr (t)= Τ cr (0)sqrt(GDP(t)/GDP(0)
M(t)=M(Tcr)exp(-α (t-Tcr)/L)=SL(1-exp(-αTcr/L)) exp(-α 1 (t-Tcr)/L)
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The model - from microeconomic approach to macroeconomic model
Tcr increases with economic growth economic growth:
(4)
(5)
Evolution of the PID in the USA: high incomes
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15. The Pareto threshold
(HIGH INCOME DEFINITION)
= MPareto(0)(GDP(t)/GDP(0))
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(6)
Defining the Pareto threshold
PERSONAL INCOME > MPareto(t) =
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16. The Pareto threshold - estimate
(observed and predicted personal income distributions for the year 1999
– a portion of population with income below a given value)
Defining the Pareto threshold
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17. The Pareto threshold - estimate
(observed and predicted personal income distributions for the year 1999
– a portion of population with income below a given value)
Defining the Pareto threshold
The curves diverge at income of $54K – the Pareto distribution threshold
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18. Observed and predicted cumulative PIDs for 1999 – a portion of total income
received by population with income below a given value. The ratio of the
observed cumulative income of the population above the Pareto threshold
(0.450) and the corresponding theoretical value (0.333) and is equal to 1.35.
Defining the Pareto threshold
This value is considered as an effective increase of the average capacity
to earn money for people above the Pareto threshold
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19. Results: Distribution above $100K
Results from modelling: the PID in the USA is predefined
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20. Results: Distribution above the Pareto threshold
Results from modelling: the PID in the USA is predefined
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Evolution of the PID in the USA: high incomes
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21. Distribution above the Pareto threshold: from 2002 to 2023
Results from modelling: the PID in the USA is predefined
Per capita real GDP growth rate = 1.6% per year
Population projection - The US Census Bureau
http://www.census.gov/population/www/projections/popproj.html
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22. Conclusions (1)
1. In the high-income range, the PID in the USA is represented by a
power law (Pareto) distribution
2. The Pareto distribution results from self-organized criticality (SOC)
3. The SOC effectively increases the capacity to earn money
4. A macroeconomic model is developed to predict the PID shape
and the PID evolution, including the high-income range
5. The model predicts the observed dependence of the PID on work
experience in the high-income range
6. The model predicts evolution of the dependence with time
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23. Conclusions (2)
7. US society has structured personal incomes in a fixed manner
8. There is no economic means to change the PID: only social
structure (ranking of the capability to earn money and size of
earning means) produces the observed inequality
9. Socialism economically is less effective than capitalism due to the
personal income limit
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