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The Dynamics of Personal Income Distribution andInequality in the United States
Oleg I. Kitov 1 and Ivan O. Kitov 2
5th ECINEQ MeetingBari, 22 July 2013
1Department of Economics and Institute for New Economic Thinking at the OxfordMartin School, University of Oxford
2Russian Academy of SciencesO.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 1 / 24
Overview
We model personal income dynamics, personal income distribution andinequality in the United States. Note that the measurement unit is anindividual, as opposed to a tax unit or a household. The structure of thepresentation 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
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
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
Model: Overview
Two-factor model for the evolution of individual money income with workexperience. 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 pointand then start declining.
2 Super-critical region: if personal income reaches a certain (Pareto)threshold, it does not follow the two-factor model dynamics, butrather a follows power law, i.e. a person can obtain any income withrapidly decreasing probability.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 5 / 24
Model: Personal Income Growth
The rate of change of personal money income, M(t), for an individualwith work experience t is modeled as a dissipation process that depends ontwo 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 themodified ability to earn money.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 6 / 24
Model: Time Dependence of Parameters
Ability to each money and the instrument are allowed to vary withexperience, t, and calendar years, τ . For simplicity we assume that bothevolve 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 withgrowth of real GDP per capita. We call this product a capacity to earnmoney.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 7 / 24
Model: Relative Parameters
We assume that Σ (τ0, t) and Λ (τ0, t) are bounded above and below andintroduce the corresponding dimensionless variables, which are measuredrelatively 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
Model: Distribution of Parameters
We allow the relative initial values of S (τ0, t0) and L (τ0, t0), for any τ0and t0, to take discrete values from a sequence of integer numbers rangingfrom 2 to 30, with uniform probability distribution over realizations. Therelative capacity for a person to earn money is distributed over the workingage 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 SiLj define a unique time history ofincome rate dynamics. No individual future income trajectory is predefined,but it can only be chosen from the set of 841 predefined individual pathsfor 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
Model: Solution with Constant Parameters
For simplicity, assume that ability and instrument parameters do notchange over time and solve the model analytically. The solution is:
M(t) = ΣΛ(
1− exp(−α
Λt))
(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
Model: Solution ContinuedSubstitute in the product of the relative values Si and Lj , time dependentminimum values Σmin and Λmin, and normalize to the maximum valuesΣmax , and Λmax , to get Mij (t):
M̃ij (t) = ΣminΛminS̃i L̃j
(1− exp
{−(
1
Λmin
)(α̃
L̃j
)t
})(8)
where
Mij (t) =Mij (t)
SmaxLmax
S̃i =Si
Smax
L̃i =Li
Lmax
α̃ =α
Lmax
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 11 / 24
Model: Simulated Income Growth Paths
0 20 40 600
0.2
0.4
0.6
0.8
1
Normalizedincome
1930
2x2, model2x2, real30x30, model30x30, real
0 20 40 600
0.2
0.4
0.6
0.8
1
Normalizedincome
2011
2x2, model2x2, real30x30, model30x30, real
The evolution of personal income for different capacities for a 75-year-oldperson (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
Model: Income Decay
Average income among the population reaches its peak at some age andthen starts declining. In our model, we set the money earning capability tozero Σ (t) = 0, at some critical at some critical work experience, t = Tc .The solution for t > Tc is:
M̃ij (t) = ΣminΛminS̃i L̃j
(1− exp
{−(
1
Λmin
)(α̃
L̃j
)Tc
})× (9)
exp
{−(
1
Λmin
)(γ̃
L̃j
)(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 ratefor work experience above Tc .
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 13 / 24
Model: Simulated Income Paths with Decay
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Normalizedincome
1930: 2x21930:30x302011: 2x22011:30x30
The change in the personal income distribution between 1930 and 2011associated with growing Tc and larger earning tool, L. The exponential fallafter Tc is taken into account.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 14 / 24
Pareto Distribution
In order to account for top incomes, which evolve according to a powerlaw, we need to assume that there exists some critical level of income ratethat separates the two income classes: exponential and Pareto. We willrefer to this level as Pareto threshold income, Mp (t). Any person reachingthe Pareto threshold can obtain any income in the distribution with arapidly decreasing probability governed by a power law. Pareto threshold isevolving 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 andobtain an opportunity to get rich.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 15 / 24
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 inincome 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 andMaddison.
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
Data: CPS Tabulated Age-dependent Incomes
1950 1960 1970 1980 1990 2000 20100.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Pro
portionofpopulationwithincome
Working age15-2425-3435-4445-5455-64
1950 1960 1970 1980 1990 2000 20100
0.025
0.05
0.075
0.1
0.125
Pro
portionofpopulationin
theopen-endedbin
Working age15-2425-3435-4445-5455-64
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 beenfalling since 1979. Notice the break in the distributions between 1977and 1979 induced by large revisions implemented in 1980.
2 Right panel: The portion of population in the open-end high incomebin.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 17 / 24
Data: Normalized Age-dependent Mean Income
10 20 30 40 50 60 70 800
1
2
3
4
5
6x 10
4M
eanIncome,2011dollars
1948196819882011ObservedPeak
1950 1960 1970 1980 1990 2000 20100
0.2
0.4
0.6
0.8
1
Norm
alizedincome,2011dollars
Working age15-2425-3435-4445-5455-64
1 Left panel: years 1948, 1960, 1974, 1987, and 2011- mean incomenormalized to peak value in these years.
2 Right panel: Mean income in various age groups normalized to peakvalue in a given year. The age of peak mean income changes withtime.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 18 / 24
Calibration and Model Predictions
1 Assume every individual in the population starts earning income atthe age of 15.
2 For each year, τ , and work experience, t, calibrate model parametersto the tabulated age-dependent CPS data, GDP per capita growthand age-distribution of the population:
I Initial values of Σ and ΛI critical age Tc
I exponents α and γI Pareto threshold Mp
3 Predict incomes for each individual in a given year, depending onhis/her work experience, based on the assumption of uniformdistribution of ability and instrument over every year of age.
4 Aggregate individual incomes over work experience cohorts to matchthe 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
Results: Predicting Mean Income
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
1.2
Normalizedmeanincome
Model incomeActual income
When we aggregate the model and estimate the (normalized) meanincomes in the 5-year age bins we can predict the actual data availablefrom CPS quite accurately. Figure compares model predictions with actualmean incomes in 1998.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 20 / 24
Results: Predicting Overall Gini
1940 1950 1960 1970 1980 1990 2000 20100.45
0.5
0.55
0.6
0.65
0.7
Gini
Working age
ModelCPS with incomeCPS all
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 21 / 24
Results: Predicting Age-dependent Gini
1940 1960 1980 2000 20200.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 25-34
ModelCPS with incomeCPS all
1940 1960 1980 2000 20200.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 35-44
ModelCPS with incomeCPS all
1940 1960 1980 2000 20200.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 45-54
ModelCPS with incomeCPS all
1940 1960 1980 2000 20200.35
0.4
0.45
0.5
0.55
0.6
0.65
Gini
Age 55-64
ModelCPS with incomeCPS all
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 22 / 24
Results: Predicting Age-dependent Gini
1 For the age groups with high ratios of individuals with income, 45-55and 54-65, the three curves are much closer.
2 The two observed curves should converge as the proportion of peoplewith income approaches unity, which should also result in our modelpredictions matching the observations much closer.
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 23 / 24
THANK YOU!
O.I. Kitov and I.O. Kitov Personal Income Distrubution in the U.S. ECINEQ, Bari 2013 24 / 24