The dynamics of personal income distribution and inequality in the United States

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


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.


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.


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.


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.


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)


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.


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


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


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.


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 .


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.


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.


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


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.


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.


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.


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.


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


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


1940 1960 1980 2000 20200.35

0.4

0.45

0.5

0.55

0.6

0.65

Gini

Age 35-44


1940 1960 1980 2000 20200.35

0.4

0.45

0.5

0.55

0.6

0.65

Gini

Age 45-54


1940 1960 1980 2000 20200.35

0.4

0.45

0.5

0.55

0.6

0.65

Gini

Age 55-64



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.


THANK YOU!


The dynamics of personal income distribution and inequality in the United States

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