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, Tc, 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.
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, 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.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.