We model the evolution of age-dependent personal income distribution and inequality as expressed by the Gini ratio. In our framework, inequality is an emergent property of a theoretical model we develop for the dynamics of individual incomes. The model relates the evolution of personal income to the individual’s capability to earn money, the size of her work instrument, her work experience and aggregate output growth. Our model is calibrated to the single-year population cohorts as well as the personal incomes data in 10- and 5- year age bins available from March Current Population Survey (CPS). We predict the dynamics of personal incomes for every single person in the working-age population in the USA between 1930 and 2011. The model output is then aggregated to construct annual age-dependent and overall personal income distributions (PID) and to compute the Gini ratios. The latter are predicted very accurately - up to 3 decimal places. We show that Gini for people with income is approximately constant since 1930, which is confirmed empirically. Because of the increasing proportion of people with income between 1947 and 1999, the overall Gini reveals a tendency to decline slightly with time. The age-dependent Gini ratios have different trends. For example, the group between 55 and 64 years of age does not demonstrate any decline in the Gini ratio since 2000. In the youngest age group (from 15 to 24 years), however, the level of income inequality increases with time. We also find that in the latter cohort the average income decreases relatively to the age group with the highest mean income. Consequently, each year it is becoming progressively harder for young people to earn a proportional share of the overall income.

1. The Dynamics of Personal Income Distribution and
Inequality in the United States
Oleg I. Kitov 1 and Ivan O. Kitov 2
5th ECINEQ Meeting
Bari, 22 July 2013
1
Department of Economics and Institute for New Economic Thinking at the Oxford
Martin School, University of Oxford
2
Russian Academy of Sciences
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 1 / 24

2. Overview
We model personal income dynamics, personal income distribution and
inequality in the United States. Note that the measurement unit is an
individual, as opposed to a tax unit or a household. The structure of the
presentation is as follows:
1 Motivation: Inequality through personal income dynamics.
2 Model: An equation for person incomes dynamics.
3 Data: Age-dependent incomes and inequality from tabulated CPS.
4 Calibration: Model parameters implied by CPS data.
5 Results: Age-dependent Gini indices predicted by the model.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 2 / 24

3. Motivation: Micro Level
Differences in Personal Incomes
Growing income variance within a cohort as it ages.
Disproportionately high top income shares.
Long tails and skewness to the right.
Lower median earning relatively to the mean.
Varying peak income age for different education levels.
Growing age of peak mean income.
Falling share of income attributed to youngest cohorts
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 3 / 24

4. Motivation: Macro Level
Measurements of Income Distribution and Inequality
Parametrized distributions: logarithmic, exponential, gamma etc.
Top incomes satisfying power law distribution
Increasing top income shares using IRS data (Piketty and Saez, 2003).
No increase using CPS data (Burkhauser et at, 2012).
Age-dependent distributions not modeled.
Macro measures not reconciled with micro facts and theory.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 4 / 24

5. Model: Overview
Two-factor model for the evolution of individual money income with work
experience. Two income regions governed by distinct laws:
1 Sub-critical region: a two factor model for personal income growth.
Incomes grow with work experience, reach a peak at a certain point
and then start declining.
2 Super-critical region: if personal income reaches a certain (Pareto)
threshold, it does not follow the two-factor model dynamics, but
rather a follows power law, i.e. a person can obtain any income with
rapidly decreasing probability.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 5 / 24

6. Model: Personal Income Growth
The rate of change of personal money income, M(t), for an individual
with work experience t is modeled as a dissipation process that depends on
two indepenendent parameters (latent factors):
1 Ability to earn money (human capital): σ (t)
2 Instrument for earning money (job type): Λ (t)
The differential equation for the evolution of personal income is given by:
dM(t)
dt
= σ(t) −
α
Λ(t)
M(t) (1)
where α is the dissipation coefficient. We also introduce a time dimension,
τ, which represents calendar years. Finally, let Σ (t) = σ(t)
α be the
modified ability to earn money.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 6 / 24

7. Model: Time Dependence of Parameters
Ability to each money and the instrument are allowed to vary with
experience, t, and calendar years, τ. For simplicity we assume that both
evolve as a square root of aggregate output per capita, Y (t):
Σ (τ0, t) = Σ (τ0, t0)
Y (τ)
Y (τ0)
= Σ (τ0, t0)
Y (τ0 + t)
Y (τ0)
(2)
Λ (τ0, t) = Λ (τ0, t0)
Y (τ)
Y (τ0)
= Λ (τ0, t0)
Y (τ0 + t)
Y (τ0)
(3)
Note that the product Σ (τ0, t0) Λ (τ0, t0) evolves with time in line with
growth of real GDP per capita. We call this product a capacity to earn
money.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 7 / 24

8. Model: Relative Parameters
We assume that Σ (τ0, t) and Λ (τ0, t) are bounded above and below and
introduce the corresponding dimensionless variables, which are measured
relatively to a person with the minimum values:
S (τ0, t) =
Σ (τ0, t)
Σmin (τ0, t)
(4)
L (τ0, t) =
Λ (τ0, t)
Λmin (τ0, t)
(5)
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 8 / 24

9. Model: Distribution of Parameters
We allow the relative initial values of S (τ0, t0) and L (τ0, t0), for any τ0
and t0, to take discrete values from a sequence of integer numbers ranging
from 2 to 30, with uniform probability distribution over realizations. The
relative capacity for a person to earn money is distributed over the working
age population as the product of independently distributed Si and Lj :
Si (τ0, t) Lj (τ0, t) ∈
2 × 2
900
, . . .
2 × 30
900
,
3 × 2
900
, . . . ,
30 × 30
900
(6)
Each of the 841 combinations of Si Lj define a unique time history of
income rate dynamics. No individual future income trajectory is predefined,
but it can only be chosen from the set of 841 predefined individual paths
for each single year of birth, or equivalently initial work year τ0.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 9 / 24

10. Model: Solution with Constant Parameters
For simplicity, assume that ability and instrument parameters do not
change over time and solve the model analytically. The solution is:
M(t) = ΣΛ 1 − exp −
α
Λ
t (7)
Personal income rate is an exponential function of:
1 Work Experience, t
2 Capability to earn money, Σ
3 Instrument to earn money, Λ
4 Output per capita growth, Y
5 Dissipation rate, α
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 10 / 24

11. Model: Solution Continued
Substitute in the product of the relative values Si and Lj , time dependent
minimum values Σmin and Λmin, and normalize to the maximum values
Σmax , and Λmax , to get Mij (t):
˜Mij (t) = ΣminΛmin
˜Si
˜Lj 1 − exp −
1
Λmin
˜α
˜Lj
t (8)
where
Mij (t) =
Mij (t)
Smax Lmax
˜Si =
Si
Smax
˜Li =
Li
Lmax
˜α =
α
Lmax
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 11 / 24

12. Model: Simulated Income Growth Paths
0 20 40 60
0
0.2
0.4
0.6
0.8
1
Normalizedincome
1930
2x2, model
2x2, real
30x30, model
30x30, real
0 20 40 60
0
0.2
0.4
0.6
0.8
1
Normalizedincome
2011
2x2, model
2x2, real
30x30, model
30x30, real
The evolution of personal income for different capacities for a 75-year-old
person (work experience 60 years) in 1930 and 2011.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 12 / 24

13. Model: Income Decay
Average income among the population reaches its peak at some age and
then starts declining. In our model, we set the money earning capability to
zero Σ (t) = 0, at some critical at some critical work experience, t = Tc.
The solution for t > Tc is:
˜Mij (t) = ΣminΛmin
˜Si
˜Lj 1 − exp −
1
Λmin
˜α
˜Lj
Tc × (9)
exp −
1
Λmin
˜γ
˜Lj
(t − Tc)
1 The first term is the level of income rate attained at time Tc.
2 The second term represents an exponential decay of the income rate
for work experience above Tc.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 13 / 24

14. Model: Simulated Income Paths with Decay
0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
Normalizedincome
1930: 2x2
1930:30x30
2011: 2x2
2011:30x30
The change in the personal income distribution between 1930 and 2011
associated with growing Tc and larger earning tool, L. The exponential fall
after Tc is taken into account.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 14 / 24

15. Pareto Distribution
In order to account for top incomes, which evolve according to a power
law, we need to assume that there exists some critical level of income rate
that separates the two income classes: exponential and Pareto. We will
refer to this level as Pareto threshold income, Mp (t). Any person reaching
the Pareto threshold can obtain any income in the distribution with a
rapidly decreasing probability governed by a power law. Pareto threshold is
evolving in time according to:
Mp (τ) = Mp (τ0)
Y (τ)
Y (τ0)
(10)
People with high enough Si and Lj can eventually reach the threshold and
obtain an opportunity to get rich.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 15 / 24

16. Data: CPS Age-dependent Incomes
We use tabulated CPS data from 1947 to 2011.
Age-dependent data is available in 5- and 10-year age cohorts in
income bins.
The number and width of income bins have been revised several times.
Data distorted by top-coding of incomes.
Annual output per capita growth rates are taken from BEA and
Maddison.
Data on population age distribution is taken from the Census Bureau
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 16 / 24

17. Data: CPS Tabulated Age-dependent Incomes
1950 1960 1970 1980 1990 2000 2010
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Proportionofpopulationwithincome
Working age
15-24
25-34
35-44
45-54
55-64
1950 1960 1970 1980 1990 2000 2010
0
0.025
0.05
0.075
0.1
0.125
Proportionofpopulationintheopen-endedbin
Working age
15-24
25-34
35-44
45-54
55-64
1 Left panel: Portion of population with income in various age groups.
In the group between 15 and 24 years of age, the portion has been
falling since 1979. Notice the break in the distributions between 1977
and 1979 induced by large revisions implemented in 1980.
2 Right panel: The portion of population in the open-end high income
bin.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 17 / 24

18. Data: Normalized Age-dependent Mean Income
10 20 30 40 50 60 70 80
0
1
2
3
4
5
6
x 10
4
MeanIncome,2011dollars
1948
1968
1988
2011
Observed
Peak
1950 1960 1970 1980 1990 2000 2010
0
0.2
0.4
0.6
0.8
1
Normalizedincome,2011dollars
Working age
15-24
25-34
35-44
45-54
55-64
1 Left panel: years 1948, 1960, 1974, 1987, and 2011- mean income
normalized to peak value in these years.
2 Right panel: Mean income in various age groups normalized to peak
value in a given year. The age of peak mean income changes with
time.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 18 / 24

19. Calibration and Model Predictions
1 Assume every individual in the population starts earning income at
the age of 15.
2 For each year, τ, and work experience, t, calibrate model parameters
to the tabulated age-dependent CPS data, GDP per capita growth
and age-distribution of the population:
Initial values of Σ and Λ
critical age Tc
exponents α and γ
Pareto threshold Mp
3 Predict incomes for each individual in a given year, depending on
his/her work experience, based on the assumption of uniform
distribution of ability and instrument over every year of age.
4 Aggregate individual incomes over work experience cohorts to match
the format of 5- and 10-year age cohorts in the tabulated CPS data.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 19 / 24

20. Results: Predicting Mean Income
0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
1.2
Normalizedmeanincome
Model income
Actual income
When we aggregate the model and estimate the (normalized) mean
incomes in the 5-year age bins we can predict the actual data available
from CPS quite accurately. Figure compares model predictions with actual
mean incomes in 1998.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 20 / 24

21. Results: Predicting Overall Gini
1940 1950 1960 1970 1980 1990 2000 2010
0.45
0.5
0.55
0.6
0.65
0.7
Gini Working age
Model
CPS with income
CPS all
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 21 / 24

22. Results: Predicting Age-dependent Gini
1940 1960 1980 2000 2020
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 25-34
Model
CPS with income
CPS all
1940 1960 1980 2000 2020
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 35-44
Model
CPS with income
CPS all
1940 1960 1980 2000 2020
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 45-54
Model
CPS with income
CPS all
1940 1960 1980 2000 2020
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 55-64
Model
CPS with income
CPS all
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 22 / 24

23. Results: Predicting Age-dependent Gini
1 For the age groups with high ratios of individuals with income, 45-55
and 54-65, the three curves are much closer.
2 The two observed curves should converge as the proportion of people
with income approaches unity, which should also result in our model
predictions matching the observations much closer.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 23 / 24

24. THANK YOU!
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 24 / 24