1.

**Physical intuition behind income growth and fall**

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 include paid job, government transfers, bank interest,
capital gain, inter-family transfers, and others. The U.S. Census Bureau questionnaire
[2006] lists 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 Integrated Public Use
Microdata Series (IPUMS) preparing and distributing data for the broader research
community [King *et al*., 2010]. 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 of 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 a 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 on 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 cells [IRS, 2015], 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]. Nevertheless,
between 40 and 60 years of age, all curves in the right panel of 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. The 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 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 entire 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 different if the IRS sources are included and some CPS
income sources are excluded [Henry and Day, 2015, Ruser *et al*., 2004; Weinberg, 2004; U.S. Census Bureau, 2015b]. 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 personal income inequality based on the
IRS data exclude half of the population, the poorest half. It is difficult to
consider such estimates as accurate and helpful for understanding the
mechanisms of the income distribution. The BEA income data are worthless for
quantitative analysis of individual incomes - no age, gender, race information
is available.

Astoundingly, the principal features observed
in Figure 1 can be accurately approximated by basic mathematical functions.
Moreover, these functions represent solutions of simple ordinary differential
equations. The 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-*18*)*)] + 0.09, where *t* is the age. The overall fit between the measured and
approximating curves is extremely good from 18 to 55 years of age before the
mean income curve starts to fall.

The approximating equation is a
well-known function often called the “exponential saturation function”. This
function represents a closed-form solution of a simple ordinary differential
equation dx(*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. The 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*)<X_{max},
*t* →∞.

A standard example in general physics to
illustrate the saturation process is associated with the heating of a metal ball by
an internal source with constant power, U. The growth in temperature, *T*, is balanced by energy loss through
the 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* _{v}*, one can write the
following equation:

4/3π*R*^{3}C* _{v}*d

*T*(

*t*)/d

*t*= U –

*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 of (1) by 4/3π*R*^{3}C* _{v}* we obtain:

d*T*(*t*)/d*t*
= Ũ*
*– *D̃*T(*t*)/*R* (2)

where
Ũ=3U/(C* _{v}*4π

*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}
+ (Ũ*R*/*D̃*)[1 - exp(-*D̃t*/*R*)]
(3)

Relationship (3) implies that
temperature approaches its maximum value Ũ*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 a better understanding of our model and
income observations. We interpret temperature as income, which one can reach
using some physical capital, say, 4/3πR^{3}, and personal efforts, say,
U. 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. The 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 the hard sciences. The exponential function is a
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. 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 U=0 in (1).

Hence, the observed features of the mean
income behaviour are similar to those observed in simple physical experiments.
However, we need to describe the income trajectory for each and every person in a
given economy. It is natural to suggest that all individual incomes follow their own
saturation curves and their average value follows up some individual
trajectory. Then the distribution of parameters defining individual
trajectories, *i.e*. income analogues
of R and U, is completely constrained by observations. This is the intuition
behind our microeconomic 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 the heated sphere of radius R,
the sizes of inhomogeneous inclusions are distributed according to a power-law *L*^{3}dn/d(ln*L*) =const, where n is the number of inclusion of size *L* in a unit volume. This distribution
defines the structural self-similarity of fractals.

Let us consider that deformation starts
at time *t*_{0} and all
stresses are zero before. Then stresses will rise at different rates for
different inclusion sizes. At time *t*,
there is some inclusion with size *L*_{M},
which reaches its highest possible stress balancing deformation and
dissipation. At all bigger inclusions,
stress is still growing. When the rate of deformation is high enough and there
are big enough inclusions the attained stress may exceed at some point the
critical stress of fracturing. Then a quake may occur. This is a transition to
a super-critical regime and the sizes of earthquakes are distributed by a power
law.

In economics, higher incomes are characterized by a similar distribution, but they are the net result of all forces and agents in the economy, which both vary with time. They do not represent a 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 exponential distribution of low-middle incomes 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 Subsections.

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