Here,
a microeconomic model is presented, which has been developed to quantitatively
describe the dynamics of personal income growth and distribution [Kitov,
2005a]. The model is based on one principal assumption that each and every
individual above fifteen years of age has a personal capability to work. In
essence, the capability to work is equivalent to the capability to earn money.
To get money income, individuals have to use one or several means or tools from
the full set of options that may includes paid job, government transfers, bank
interest, capital gain, inter-family transfers, and others. The U.S. Census
Bureau questionnaire [2006] includes tens of money income components. It is
important to stress that some principle sources of income are not included in
the CB definition, which results in the observed discrepancy between aggregate (gross) personal income (GPI), as
reported by the Bureau of Economic Analysis and the gross money income
calculated by the CB.

In this section we summarize the formulation
of a theoretical model, originally described in Kitov [2005a]

**,**and present it as a closed form solution in a simplified setting. Figure 1 illustrates a few general features any consistent model has to describe quantitatively. In the left panel, we display the evolution of mean income curves from 1962 to 2013. The original income data are borrowed from the IPUMS [2015]. These are income microdata,*i.e.*each and every person from the IPUMS tables is characterized (among other features) by age, gender, race, gross income, and the population weight, which allows projection the individuals from the CPS population universe to the entire population. Using age, income and population weight we have calculated the age dependent mean income for all years and then normalized them to their respective peak values. The normalized curves better illustrate the growth in the age of peak income – from below 40 in the earlier 1960s to 55 in the 2010s. This is a sizeable change likely expressing the work of inherent mechanisms driving the evolution of personal income distribution. One cannot neglect the effect of increasing age when people reach their peak incomes – neither from theoretical nor from the practical point of view.
In
the right panel of Figure 1, we compare various mean income curves reported by
two different organizations responsible for income measurements: the Census
Bureau (CPS) and the Internal Revenue Service (IRS). The latter organization
does not publish the age distribution of income at a regular basis and only the
year of 1998 is available for such a comparison. The IRS mean income is
calculated in 5-year age bins, the CPS prepares historical datasets with a
5-year granularity since 1993, and the annual estimates are available from the
IPUMS microdata. The annual curve has also been smoothed with a nine-year
moving average, MA(9). As in the left panel, all curves are normalized to their
peak values.

There
are significant differences in income sources and population coverage used by
the CPS and IRS [Kitov, 2014]. Between 40 and 60 years of age, all curves in
Figure 1 are close to each other. With
regard to
the age of peak income, the CPS and IRS give identical results to the extent
the age aggregation allows. The IPUMS curve has been smoothed and thus might
have a slightly biased peak age. Between 25 and 40 years of age, the difference
in normalized mean income is larger likely because of the difference in income
sources. Same effect is observed in the eldest age groups, where taxable
incomes are not so often and the CPS curve is above the IRS one.

The
closeness of the peak ages measured by the IRS and CPS is important for our
consideration of the model applicability and reliability. The accuracy of
income measurements, the coverage of population and income source, the
level of historical consistency in
income definition and survey methodology, the diversity of personal
characteristics, and the length of time series provided by the Census Bureau
all these features make it inevitable to use the CPS data for quantitative modelling. The reverse side of this choice is the necessity to defend the
modelling results against the accusation
that the CPS data are not full and representative.

It
is true that the CPS misses some important sources of higher incomes, but
Figure 1 stresses that the estimates of key features are not be different if
the IRS sources are included. Besides, the CB provides the best income
estimates for the poorest population, where incomes are just several dollars
per year. Other organizations ignore small incomes. As a result, the estimates
of income inequality based on the IRS data exclude half of population, the
poorest half. It is difficult to consider such estimates as accurate and
helpful for understanding the mechanisms of income distribution. The BEA income
data are absolutely worthless for quantitative analysis - no age, gender, race
information is available.

Figure
1. Left panel: The change in the shape of mean income dependence on age from
1962 to 2013 as measured by the Census Bureau in the March Supplements of the
Current Population Survey. All curves are normalized to their respective peak
values. Right panel: Comparison of mean income dependence on age as measured by
the Census Bureau (CPS) and the Internal Revenue Service (IRS). The only year
with data available from the IRS is 1998.

It
is very important to stress that the features observed in Figure 1 can be
approximates by simple mathematical functions. Moreover, these functions represent
solutions of simple ordinary differential equations. Solid red line in the
right panel is calculated to fit the CPS mean income curve. For this line, the
equation is [1 -

*exp*(-0.071(*t-*15*)*)] + 0.09, where*t*is the age. The overall fit between the measured and approximating curves is extremely good between 18 and 55 years of age, before the mean income curve starts to fall.
The
approximating equation is a well-known function often called “exponential
saturation function”. This function represents a closed-form solution of a
simple ordinary differential equation dx(max

*t*)/d*t*=*a*-*b*x(*t*), where*a>*0 and*b>*0 are constants. The match between the observed and approximating curves provides some hint on the forces behind income growth. Second term in the above equation represents the force counteracting the unlimited growth of x(*t*). The amplitude of the counteracting force is proportional to the attained level, and that implies the finite value of x(*t*)*t*→∞.

A
standard example in general physics to illustrate the saturation process is
associated with heating of a metal ball by an internal source with constant
power, W. The growth in temperature,

*T*, is balanced by energy loss through the ball surface, and the energy flux through the surface is proportional to the attained temperature. Thermal conductivity can be treated as infinite in terms of the characteristic time of all other processes. For a ball of radius*R*and volumetric heat capacity, C*, one can write the following equation:*_{v}
4/3π

*R*^{3}C*d*_{v}*T*(*t*)/d*t*= W –*DT*(*t*)4π*R*^{2}(1)
where

*D*is a constant defining the efficiency of heat loss through the surface, which is similar to dissipation. By dividing both sides if (1) by 4/3π*R*^{3}C*we obtain:*_{v}
d

*T*(*t*)/d*t*= W̃*–**D̃*T(*t*)/*R*(2)
where
W̃=3W/(C

*4π*_{v}*R*^{3}) is the specific power of the heating sources expressed in units of thermal capacity, and*D̃ =*3*D/C*The solution of (2) is as follows:_{v}.*T*(

*t*) =

*T*

_{0}+ (W̃

*R*/

*D̃*)[1 - exp(-

*D̃t*/

*R*)] (3)

Relationship
(3) implies that temperature approaches its maximum value W̃

*R*/*D̃*along the saturation trajectory, which we also observe in Figure 1. Instructively, the maximum possible temperature is proportional to*R*. This fact is helpful and important for better understanding of our model. We interpret temperature as income, which one can reach using some physical capital, say 4/3πR^{3}, and personal efforts, say W. Then the saturation curve in Figure 1 becomes an obvious result.
Above
the age of peak mean income in Figure 1, one observes an exponential fall. Blue
dotted line is defined by function

*exp*[-0.052(*t*-56)]. It best matches the IRS curve above 56 years of age. The match between the observed curve and the exponent is extraordinary even in terms of hard sciences. The exponential function is a well-known solution of a familiar equation: dx(*t*)/d*t*=-*b*x(*t*). The only difference is in the absence of term*a*, but now the curve starts from 1.0. Hence, the evolution of mean income measured by the IRS above the critical age can be expressed by a differential equation formally identical to that describing free cooling of a preheated sphere,*i.e.*when heating source W=0 in (1).
Hence,
the observed features of the mean income behaviour are similar to those
observed in a simple physical experiment. However, we need to describe income
trajectory for each and every person in a given economy. It is natural to
suggest that all individual incomes follow own saturation curves and their
average value repeats some individual trajectory. Then the distribution of
parameters defining individual trajectories,

*i.e*. income analogues of R and W, is completely constrained by observations. This is the intuition behind our model.
Originally,
the idea of income modelling with equation (2) came from geomechanics [Rodionov

*et al*., 1982]. An identical equation describes the growth of stress, σ(*t*), in an inhomogeneous inclusion with characteristic size L experiencing deformation at a constant rate ε̇ as induced by external forces. Solution (3) is important to predict the highest possible level of stress at a given inclusion with size L. Unlike in the simple experiment with heated sphere of radius R, the sizes of inhomogeneous inclusions are distributed according to a power law L^{3}dn/d(lnL) =const, where n is the number of inclusion of size L in a unit volume. This distribution defines a structural self-similarity of fractals.
In
economics, higher incomes are characterized by a similar distribution, but they
are the net result of all forces and agents in the economy not the predefined
structure as in geomechanics. Moreover, low and middle incomes are distributed
according to an exponential law rather than a power one. So, we had to construct the basic distributions
of defining parameters which result in the exponential distribution of
low/middle income and power law distribution above the Pareto threshold. The
process of model development with explicit differential equations together with
the selection of underlying distributions is described in the following
subsection.

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