Our model stems from extensive physical intuition and is supported by
direct comparison of income observations with closed-form solutions of simple
differential equations describing fundamental physical processes. The set of
equations describing the growth and fall of incomes is fully borrowed from
physics, with the empirically estimated constant of dissipation and the
distribution of sizes of personal capabilities and instruments. The transition
to the power law distribution of the highest incomes is also a physical process,
which can be qualitatively described by the concept of self-organized
criticality. In that sense, the dynamics of the highest incomes is not governed
by simple physical relationships – the power law distribution is a purely
statistical description rather than a solution of a system of differential
equations. Nevertheless, all properties in the super-critical regime are
defined by two parameters – the number of people above the Pareto threshold and
the power law index. The former parameter is exactly predicted by our model as
a function of time and age for males and females. The index has to be
empirically estimated, as in other physical cases like for the slopes of
earthquake recurrence curves in various seismic regions [Kitov

*et al.*, 2011]. Overall, the system of personal income distribution is fully and accurately described by physical equations. In that sense, it is a physical system.
We introduced the
microeconomic model of personal income distribution a decade ago and used the
CPS historical data to calibrate all defining parameters. The March CPS reports
aggregate incomes in five-year age cells since 1993. Ten-year cells are used
between 1947 and 1992, with sporadic appearance of shorter cells for the
youngest population. The income data and the U.S. age pyramids between 1947 and
2011 published by the U.S. Census Bureau were used as they are without gender
separation. The IPUMS income microdata not only make it possible to distinguish
between males and females but also provide various estimates in one-year cells.
In this study, we use the advantage of income microdata and model two specific
age-dependent features of personal income distribution: mean income and portion
of people in the Pareto distribution. These two features are most sensitive to
the influence of time and age. A correct income distribution model must
accurately describe the dynamics of secular and age-dependent changes observed
in actual data. Any model not predicting the dynamics of actual changes should
be disregarded. Our model successfully predicts all principal changes in both
features observed between 1962 and 2014 for males and females separately.

The difference in income
dynamics demonstrated by two genders represents enormous challenge for
quantitative
modelling. A model unifying (at first glance) incompatible
results for two genders has to be parsimonious and include only parameters
common for both cases. The dynamic discrepancy between male and female incomes
has to be explained only by values of defining parameters: constants and variables controlled by
exogenous measurable forces represented by continuous time series. The evolution
of gender-dependent income features together with all changes in the difference
between them should be driven by the same driving forces. In our model, the
only force moving personal income distribution along predefined trajectories is
real economic growth as expressed by GDP per capita calculated for working age
population.

Dynamic behavior of the
difference in income distribution between males and females requires a special approach
in quantitative models of income distribution. The original version of KKM made
no difference between men and women. Here, we extend the KKM by introducing two
independent populations with different features of income distribution as
reported by the CPS and IPUMS. Since gender divides U.S. population in
approximately equal proportions over time and age the gender-related income
effects do improve the KKM predictive power upon the original version. In other
words, females have sizeable contribution to the total income. The next step to
a more precise model might be the introduction of race differences of income
distribution. The income difference between white males and black females is
much more dramatic than income difference between two sexes considered in this
study. This is a real challenge to our income distribution model.

All in all, we have demonstrated in this paper that the
refined KKM accurately explains a number of common and gender-specific features.
The principal finding of this study is that female population in the U.S. has
the same distribution of the capability to earn money (notation similar but not
equivalent to human capital) and consistently lower sizes of work instruments (work
capital) compared to those for men. The income gap between women and men has
been closing since 1960 and currently an average female has work capital making
65% of that available for an average man. It was only 45% in the 1960s. Considering the
same capability to earn money for females, one can conclude that the relatively
lower work capitals (

*e.g.,*job positions, assets, …) are controlled by external force. A fair distribution has not been achieved yet. It will likely take decades.
The relatively lower instrument sizes available for
females make the proportion of female above the Pareto threshold lower. In turn,
this effect lowers the mean income for the same age since a relatively lower
number of rich females occurs in all age groups. However, the lack of rich
women is partially compensated by the effect of lowered Pareto threshold for
females, which is most prominent in the 1960s and 1970s. The coherent increase
in the instrument size and Pareto threshold for women has been incorporated
into our model. As a result, the model accurately predicts the early growth
trajectory, which is most sensitive to the size of work instrument, and the
number of females above their own Pareto threshold. As in the original model, both
parameters increase with time as the square root of real GDP per capita. For
women, we have introduced a specific option as revealed from observations - the
relative instrument size and the Pareto threshold both follow linear time
trends with different slopes.

The female mean income shows a very specific feature –
it is practically constant during an extended period spanning the ages between
~30 and ~60. In our model, this feature results from the fast growth of all
personal incomes to their peak values, which are then retained at the same
level. The expedite rise in all incomes is induced by the lowered sizes of work
instruments available for women. In turn, the lower instruments do not allow
personal incomes to reach the Pareto threshold and there are almost no rich
women by male standards in the 1960s and 1970s. Therefore, the disparity in
work capitals affects the low-middle incomes and higher incomes together. Such
a shelf is absent in the overall mean income curve because of larger instrument
sizes available for males.

The shelf in the females’ mean income curves has also
revealed the difference between critical times for the low-middle (in physical
notation - sub-critical) and high (super-critical) incomes, the latter governed
by the Pareto distribution. Equation (13) describing the sub–critical regime is
valid from the start of work experience to the age of retirement. Then incomes
fall along an exponential trajectory described by equation (17). The actual age
of retirement varies in a narrow band between ~60 and ~65 years and is embedded
into the model as constant. The fall is described by an exponential function
with a negative index. This is a new feature of the upgraded model. In the
original model, the critical age,

*T*, was the same for low-middle and high incomes. The input of rich men in the overall PID masked the presence of the low-middle income critical age. Instructively, the mean income measured for males supports the existence of two critical ages._{c}
The refined model includes several new features not
compromising the underlying physical concept of saturation growth and the
transition from sub-critical to super-critical regime of income distribution.
The extended version of the original model accurately predicts the PID
evolution for males and females in the U.S. from 1962 to 2014, i.e. where the
IPUMS data are available. Since the GDP estimates are available from the U.S.
Bureau of Economic Analysis since 1929 we start our model in 1930 for males. Actually,
the model spans the period since 1870,

Forced
deprivation of higher job positions (work capital) is the cause of the observed
long term income inequality between male and female in the U.S. It is not only
unjust to women but has a negative effect on real economic growth. The
replacement of highly capable women with less capable men results in lower total
income, which is an equivalent to real GDP. Women have been catching up since
the 1960s and that improves the performance of the U.S. economy. It will take
decades, however, to full income equality between genders. The problem of race
income disparity will take longer time to full resolution, however. *i.e.*the year when started their work people who reached the age of 75 in 1930. For females, the start year is shifted to 1960 because of changing relative size of work instrument and Pareto threshold.

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