Personal income distribution for males and females
We first present the change in gender difference related to the distribution of personal income as measured by the Census Bureau. In Figure 9, we compare the male and female population density as obtained in 1986 and 2014. The population density is the ratio of the number of people in a given income bin and the width of this bin. It is measured in the number of people per dollar. When integrated over the entire income range, the curves in Figure 9 give the number of people of a given gender. The IPUMS income microdata data are aggregated in $1000 bins between $0 and $200,000. These income bins are likely too narrow for the 2014 curves and they oscillate over the whole income range, even after smoothing with a MA(7). The same effect is observed at higher incomes, where the number of people is too small and many income bins are just empty. The scarcity of income data at higher incomes has a negative effect of the estimates of the Pareto index.
The level of the female curve is higher at lower incomes. The female population makes approximately 52% of the total working age population, i.e. the number of men and women is approximately equal. Therefore, Figure 9 shows that a larger portion of women has lower incomes. The females’ portion of population with the highest incomes becomes smaller and smaller with growing income. The male-female difference has been likely decreasing with time. In 2014, the males’ curve is closer to the females’ one than in 1986.
The male and female curves intersect at $14,000 in 1986 and at $30,000 in 2014 and then the male curve deviates further and further from the female curve. The mean incomes is $21,822 for men and $10,741 for women in 1986, in 2014 the mean income is $53,196 and $32,588, respectively. The median incomes were $17,114, $7,610, $36,301, and $22,240, respectively. Therefore, the population density curves intersect near the mean income for females, but the intersection point was above it is 1986 and below in 2014.
Another gender dependent feature, which will be discussed later in this Section, is the income range of the Pareto distribution. In 1986, the straight lines in the log-log scale reveal the Pareto range above $40,000 for males and above $30,000 for females. This difference needs special consideration. Essentially, the lower Pareto threshold for women is a strong manifestation of disparity. The meaning of richness is different for females with income. This difference has been improving with time, however, as two curves for 2014 demonstrate.
Figure 9. Population density (persons per $) as a function of income for male and female population in 1986 and 2014. The IPUMS income data are aggregated in $1000 bins between $0 and $200,000. All curves are smoothed by a MA(7) in order to suppress high-amplitude fluctuations. The male and female curves intersect at $14,000 in 1986 and at $29,000 in 2014.
Figure 9 also reveals the effect of topcoding. In 1986, the male population in the bin between $100,000 and $101,000 is by a factor of 4 larger than that in the adjacent bins. In 2015, the same effect is observed in the bin between $150,000 and $151,000, but the factor is 10. For females, there is a deep through preceding the peak at $100,000 and above. The topcoding may introduce a sizable bias in the estimate of the power law index.
The male and female PIDs in Figure 9 represent aggregated features, and thus, mask the effects of work experience. As we discussed in Section 1, the mean income and especially the evolution of the portion above the Pareto threshold reveal strong dependence on age. The difference in the overall PIDs may also be associated with the change in personal income distribution as a function of age and calendar time, the latter parameter is actually the time series of real GDP per capita. The purpose of the following Subsections is to reveal the difference between genders as expressed in mean income and portion of people above the Pareto threshold and interpret them as linked to the defining parameters of the microeconomic model.
The evolution of real mean income (measured in 2014 US$) for males and females is presented in Figure 10. The male curve is much higher than that for females over the period between 1967 and 2014, where the CPS historical estimates are available. The males’ mean income peaks at $55,337 in 2000 and then falls to $51,119 in 2010. The females’ curve peaks at $33,397 in 2007. This difference indicates that the males’ and females’ PIDs develop independently and react to the overall economic growth in different ways.
Figure 10. The evolution of mean income (real 2014 U.S. dollars) since 1967 (Source: Census Bureau, downloaded, September 25, 2015)
The total curve is approximately in the middle between the gender-associated curves. It is closer to the males curve in the 1960s and 1970s, however. This can be only the result of the difference in the relative weight of males and females. Figure 11 demonstrates the portion of population with income as reported by the CPS from 1947. Here, we use the CPS electronic tables and the reports available from the CPS site as scans of the historical reports in PDF format. We have digitized all these reports and now they are available as electronic tables for quantitative modeling. The IPUMS data start in 1962.
The portion of people with income among men does not alter much. It was 89% in 1947 and 90% in 2014. The peak portion of 97% observed in 1979 is likely a spike induced by a new methodology of income measurements. In 1977, a total revision to the CPS questionnaire as well as measuring procedure and the whole processing pipeline was implemented. In 1980, the males’ portion fell back to 94% and hovered near this level before the 2000, when it started to fall to the current level near 90%.
The females’ curve starts from 39% in 1947 and grows as a linear function of time reaching 76% in 1977. As a result, the share of female population with income jumps to 91% in 1979. This is a fully artificial step, however, which should not be modeled or explained by actual mechanisms of income distribution as it is related to the measuring procedure only. As for males, the portion of females with income was hovering near 90% between 1979 and 2000, and then started to fall in 2001 to the current level of 83%. The difference between the males’ and females portions is approximately 6% since 2003. The ratio of female and male portions is shown by black line in Figure 11. It was growing between 1947 and 1977 and has been around 0.93 ever since 1979.
Figure 11. The portion of male and female population with income. The female curve demonstrates a longer period of linear growth between 1947 and 1975 and then jumps by from 76% in 1977 to 91% in 1979, after a new definition of income was adopted and a new questionnaire was introduced. The black dotted line shows the ratio of female and male population with income.
Open circles in Figure 12 represent the ratio of mean incomes measured for male and female population with income since 1967. It has been decreasing from 2.43 in 1967 to 1.71 in 2014.
In view of the changing portion of population with income the ratio of male and female mean income has to be corrected. Open triangles in Figure 12 present the ratio of the CPS mean incomes divided by the ratio of populations in Figure 11. In 1967, the male mean income was by a factor 3.52 larger than that measured from the female population as a whole. Our model is based on the entire working age population and thus the females PIDs and their derivatives measured in the 1969s and 1970s may be biased. It may be difficult to match observations made from one third or a half of population.
Figure 12. The ratio of male and female mean income as reported by the CPS (open circles) and that corrected to the population without income (open triangles).
The evolution of age-dependent mean income curves is best represented when the curves measured in current dollars are normalized to their peak values. Figure 13 displays the normalized mean income curves for males and females with a ten-year step since 1967. The males’ curves in the left panel are similar to those for the total population (see Figure 1), which is not a surprise in view of the men’s dominance in the number of people with income and their higher incomes. The evolution of the males’ mean income with age is also similar to that for the whole population – a short period of fast growth, which is almost linear with age, a period of saturation to the peak value at the critical age, and then all curves fall to the level between 0.3 and 0.5 at the age of 75 years. The critical age grows from around 40 years in 1967 to above 50 years in 2007. The change in the slope of the initial quasi-linear segment as well as the increase in the critical age is well described by our model. Figure 14 compares the males’ and overall mean income curves. The slope is lower and the peak age is slightly larger for males only. In terms of our model, males may use larger sizes of work instruments than females and thus than the average sizes in our original model.
Figure 13. The mean income curves for male (left panel) and female (right panel) population. All curves are normalized to their peak values. The evolution between 1967 and 2007 is illustrated by curves with a ten-year step. For the males’ curves, the slope of the initial segment is falling with time and the age of the peak value grows with time. For females, the 1967 and 1977 curves demonstrates wider shelves between 25 and 60 years of age. All curves are smoothed with MA(7) before they are normalized.
The corresponding curves for females are shown in the right panel of Figure 13. Two curves for 1967 and 1977 demonstrate wider shelves between 25 and 60 years of age. This is an unusual feature not seen in the overall and male curves as reported by the Census Bureau. A wide shelf in the mean income implies no change with age, which would be extremely hard to model considering the effect of critical age, Tc. A possible explanation of the shelf is the absence of people with incomes above the Pareto threshold, as discussed in Subsection 2.3. The evolution of incomes in the sub-critical zone suggests that they can reach their respective peaks and retain the achieved level over time before the critical age. Judging by the curves in Figure 13, the critical age for sub-critical incomes in the 1960s and 1970s is around 60 years. This value is different from the critical age measured from the portion of people with mean income as well as from the mean income for the entire population, where the input of rich males is high.
The difference between two critical ages is crucial for our model. There was no possibility to distinguish between these two critical values using only the overall PIDs. The dominance of males who have many people with incomes above the Pareto threshold (see Subsection 2.3) may mask the difference. When modelling the females PIDs and their derivatives we need to take into account the possibility of two critical ages. The age when the portion of rich people achieves its peak should be driven by real GDP per capita. The age when people with incomes below the Pareto threshold reach the relevant critical value can be constant. Alternatively, it can change with the age of retirement in the U.S. The average retirement age for men has increased from 62 to 64 over the last 20 years and for women it rose from 55 in the 1960s to 62 in 2010 [Munnell, 2011].
In the right panel of Figure 14 we compare the overall and females curves for 1967 and 2013. The slops measured from the initial segments are quite different: the females mean income grows much faster than that of the overall population in both years and in-between. Our model implies that the initial growth is fully described by equation (13) and there exists a direct link between the slope and the absolute size of instruments used to earn money. Figure 14 suggests that the sizes of instrument available for females are much smaller than those used by males. The sizes available for females were so small in the 1960s and 1970s that practically no women could get into the Pareto distribution, i.e. overcome MP=0.43 in terms of our model.
Figure 14. The evolution of mean income for male population and the overall population for 1967 and 2013. All curves are normalized to their respective maxima. The initial segments of the males curve are characterized by slightly lower slopes while the critical ages are shifted to larger ages.
Figure 15 illustrate the appearance of rich women near 1980 by comparison with the male curves for the same years. In 1977, the females’ mean income is constant between 28 and 56 years of age. In 1982, a slight peak emerges between 40 and 45 years of age, which is closer to the critical age measured from the overall curves in Section 1. This peak is stressed by a faster fall of the females’ mean income curve beyond this critical age. In 1986, the peak at 42 years of age is moderate, but it can be clearly distinguished from the males’ peak around 47 years. In 1997, the males and females curves are getting more similar and two peaks now are separated by 7 years, 52 and 45 years of age, respectively. In 2007, two curves are much closer and their peak ages differ less than before.
For all female curves in Figure 15, the initial segments of mean income growth are characterized by larger slopes that those measured for males. This observation together with a smaller age of peak mean income indicates much smaller sizes of instruments women use to get income. The difference in the size of instruments has been likely decreasing since the start of measurements (i.e. 1962). Figure 16 illustrates the evolution of the male/female mean income ratio as obtained from the curves in Figure 13. The absolute value of the ratio decreases with time from 2.8 in 1967 to 1.87 in 2007. The peak age increases from 28 in 1967 to 59 years in 2007. This transition has to be accounted for in our model, likely through changing ratio of instrument sizes available for men and women. Such a transition process was not seen when we modeled the overall data because the contribution of rich women to the mean income is small.
There is a question on the modality of instrument distribution between men and women. One possibility is that all available instruments are distributed over 29 sizes as in the original model. Men just have larger instruments from the original set and women are deprived. With time, females gradually regain their basic right to use bigger and bigger instruments from the original set. This process explains the decreasing ratio on mean incomes and convergence of the mean income curves observed in Figure 15.
Figure 15. Comparison of the male and female mean income curves in 1967, 1977, 1987, 1997, and 2007. All curves are smoothed with MA(7)).
Figure 16. The ratio of male and female mean income as a function of age. The absolute value decreases with time from 2.7 in 1967 to 1.87 in 2007. The peak age increases from 28 in 1967 to 59 years in 2007.
Another option is that there are two different sets of instrument sizes – one for male and one for females. These two sets are chiefly independent and are obtained by changing distribution of the overall work capital (like in the Cobb-Douglas production function) over working age population. These should be some market forces that may differentiate males and females relative to work capital. The history of disproportionate acquisition of “human capital”, whatever this term means, as well as gender prejudice are not excluded from the list of these forces. The relative distribution is changing over time and results in the observed convergence of the mean income curves.
Both opportunities are equivalent than we model male and female population separately. The sizes of work instruments are diminished for females by some factor, which has to fall with time in both cases. At the same time, modelling of the overall population as a sum of males and females described by independent models is simpler if we introduce two sets of instruments sizes. Instructively, the capability to earn money in the gender-dependent models is the same for men and women. All in all, two genders are equal in terms of their ability to earn money but females are deprived in terms of work capital available.