gender-dependent distribution of population with income above the Pareto threshold

As we discussed in Section 1, another aggregate variable derived from PID and sensitive to work experience is the portion of people with the highest incomes, there the highest incomes are defined as those distributed according the Pareto law. The long-term observations in the USA reveal significant increase in the age when this portion achieves its maximum value with growing GDP per capita. Figure 17 displays five curves for selected years between 1967 and 2014. Each curve represents the age-dependent ratio of the population above predefined thresholds and total population. We have selected the thresholds ($11,000 for 1967, $20,000 for 1977, $43,000 for 1990, $65,000 for 2001, and $87,000 for 2014) which provide approximately the same peak value of the respective ratio – 18%, which is accompanied by the same total portion of population above these thresholds - from 7.5% in 1977 to 9% in 2014. So, the integral share of population with income (male and female) above the Pareto threshold is retained at the level of 9%. One should also to take into account the deficit of female population with income before 1977. The age of peak value increases with time, as discussed in Section 1. All curves are bell shaped – for the youngest and eldest population participation in the highest incomes is negligible. The rate of growth at the initial stage is the highest in 1967 and the slowest in the 2000s. Beyond the critical age, two curves for 1967 and 1977 fall faster and reach 2% at the age of 70. The rate of fall in 2014 is the lowermost and the curve drops to 5% level at the age of 80. This observation should be reproduced by our model.

There are two genders in the total population which have different contributions to the total population with the highest incomes. Figure 9 suggests that the women’s participation above the Pareto threshold in the 1960s and 1970s was extremely low.  Figure 18 displays curves similar to those in Figure 17 with the same thresholds, but for two genders separately.  In 1967, the share of rich females was less than 3% of the number of women with income, which was approximately 60% of the total number of females. Since the contribution of women was so low, the male population occupied almost all positions in the high income range. Therefore, the curves in Figure 17 and the males’ curves in Figure 18 do not differ for 1967 and 1977; expect the males’ peak portion is 28%. The women’s share has been increasing since the earlier 1980s and approximately 10% of women between 35 and 65 years of age had incomes above $87,000 in 2014. These women displaced men from the top income zone and only 22% to 24% of males had incomes above the same threshold.

Figure 17.  The portion of people above the Pareto threshold for various years between 1967 and 2014. Relevant thresholds are shown in the Figure.

Figure 18. The portion of males (left panel) and females (right panel) above the Pareto threshold for various years between 1967 and 2014. Thresholds are the same as in Figure 17.

Figure 19. The ratio of male and female portions in Figure 18.

The females’ contribution is still lower and one can estimate the time when the current convergence tendency will end in equal representation. Figure 19 shows the age-dependent ratio of the curves in the left and right panels of Figure 18. In 1967, the ratio peaks at 30 years of age and achieves the level of 30 and above. In 1980, the ratio hovers near 10 and then drops to 6 to 8 in 1990.  In the 21st century, the ratio falls to 3.5 in 2001 and currently is between 2 and 3.  The trend is quasi-exponential (R2=0.98) and the extrapolated curve will reach 1.0 in 2025-2030. This is the expected time of gender equality as related to participation in the Pareto distribution. The trend may change in the future, however. To model the past numbers of females above the Pareto threshold we may adjust the relevant defining parameters to fit the exponential fall in the male/female ratio.

Figure 20. Pair-wise comparison of male and female curves for selected years between 1967 and 2014.

Figure 20 illustrates the evolution of age dependence with time and compares male and female curves. We have estimated the age-dependent curves for the portion of population above given thresholds similar to those in Figure 18 and normalized them to their peak values. Male and female curves are compared on the year-by-year basis. The thresholds in Figure 20 are lowered in order to obtain more reliable estimates with smaller fluctuations. The negative effect of the deceased thresholds is that they are now below the Pareto one for males but still above that for females. The total portion of people above these thresholds varies from 13% in 1977 to 17% in 2014. In 1967, the males curve grows at a high rate from 23 years of age, peaks at 35 years of age, and then falls to 0 at 75 years of age. The female curve grows slowly with large fluctuations and has a sharp peak at 55 years of age. In 1977, the female curve peaks at 47 years of age and has a plateau till 65 years. After a period of expedite growth in the females’ share between 1977 and 1980, we observe that the male and female curves become closer and closer from 1990 to 2014. This is only relative convergence, however. The absolute levels differ by a factor of 2.5 in 2014. Women are as efficient as men when they get in the top income percentiles. They are underrepresented however.

Figure 21. The age-dependent potion of females in 1967 and 1977 when the Pareto thresholds are decreased to the level corresponding to women, as illustrated in Figure 9. Left panel: absolute portion. Right panel: normalized to peak value and smoother with MA(7).

The discrepancy between the male and female curves in Figure 20 observed in 1967 and 1977 may be induced by the difference in the Pareto thresholds. For females, it is much lower, as Figure 9 shows. In Figure 21 we display the age-dependent portions of females with income above $6,000 in 1967 and $14,000 in 1977, instead of $9,000 and $17,000, respectively, in Figure 20. The peak values are now around 20% of females with income; it has to be reduced according to the total share with income below 70% (see Figure 11). In terms of shape, both curves are now similar to those in Figure 15, which presents the mean income curves.


Income and gender

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.

Mean income

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.


The Pareto distribution of top incomes

Our model implies that persons with the highest S and L may have income only by a factor of 225 larger than that received by persons with the smallest S and L. The exponential term in (11) includes the size of earning means growing as the square root of the real GDP per capita. As a result, it takes longer and longer time for persons with the maximum relative values S29 and L29 to reach the maximum income rate (see Figure 4), while persons with S1 and L1 reach their peak income in a few years and then retain it at the level of GDP growth. The actual ratio of the highest and lowest incomes is tens of millions, if to consider the smallest reported of $1. All in all, our microeconomic model fails to describe the highest incomes.

Fortunately, it is not necessary to quantitatively predict the distribution of the highest incomes. Physics helps us to formulate an approach, which is based on transition between two different states of one system through the point of bifurcation. The dynamics of the system before (sub-critical state) and that beyond the bifurcation point (super-critical state) are described by quite different equations. For example, the hydrostatic equation cannot describe convective motion in liquid. Hence, it would be inappropriate to expect the equation of income growth in the sub-critical (“laminar”) regime to describe the distribution of incomes in the super-critical (“turbulent”) regime. It is favorable situation for our approach based on physical understanding of economy that the sub-critical dynamics can exactly predict the portion of system in critical state near the bifurcation point and the time of transition. For personal incomes, the point of transition is equivalent to some threshold, which separates sub- and super-critical regimes of income distribution.

So, in order to account for top incomes, which are distributed according to a power law, we assume that there exists some critical level of income that separates the two income regimes:  the exponential (sub-critical) and the Pareto one (super-critical). We call this level “the Pareto threshold”, MP(τ). Below this threshold, in the sub-critical income zone, personal income distribution (PID) is accurately predicted by our model for the evolution of individual incomes. Above the Pareto threshold, in the super-critical income zone, the observed PID is best approximated by a power law. Any person reaching the Pareto threshold can obtain any income in the distribution with a rapidly decreasing probability governed by a power law. To completely define the Pareto distribution, the model for the sub-critical zone has to predict the number (or portion) of people above the Pareto threshold, which must be in the range described by the model.  The predictive power of the model is determined by the possibility to accurately describe the dependence of the portion of people above MP on age as well as the evolution of this dependence over time. If the portion of people above the Pareto threshold fits observations then the contribution of the PID in the super-critical zone to any aggregate or disaggregate measure of personal income is completely defined by the empirically estimated power law exponent.

The mechanisms driving the power law distribution and defining the threshold are not well understood not only in economics but also in physics for similar transitions. The absence of explicit description of the driving mechanisms does not prohibit using well-established empirical properties of the Pareto distribution in the U.S. – the constancy of the measured exponential index over time and the evolution of the threshold in sync with the cumulative value of real GDP per capita [Piketty and Saez, 2003; Yakovenko, 2003; Kitov, 2005b, 2006]. Therefore, we include the Pareto distribution with empirically determined parameters in our model for the description of the PID above the Pareto threshold. The stability and accuracy of the observed power law distribution of incomes implies that we do not need to follow each and every individual income as we did in the sub-critical income zone.

The initial dimensionless Pareto threshold is found to be MP(τ0)=0.43 [Kitov, 2005a], which is within the range described by the model. Without loss of generality, we can define the initial real GDP per capita as 1. In this case, MP(τ0)=0.43 for any starting year, where Y(τ0)=1.   Then the Pareto threshold evolves with time proportionally to growth in real output per capita:

MP (τ) = MP(τ0) [Y (τ) / Y (τ0)]  = MP(τ0) Y (τ)                                                                       (20)

This retains the portion above this threshold almost constant over time as shown in Figure 7. In the model, the Pareto threshold does not depend on age.

Figure 7. The portion of people above the dimensionless Pareto threshold MP=0.43 between 1930 and 2011. The portion drops during WWII and hovers around 10% ever since.

As we discuss in the next Section, the Pareto threshold was different for males and females in the 1960s and 1970s. This difference is one of principal features of the general gender disparity in the U.S. and deserves deeper analysis and special modelling. It is easier to incorporate the observed lower Pareto threshold for women in our model than to understand the forces behind such a difference. In this Section, we discuss the overall model and illustrate it by income features of the total population.

Theoretically, the cumulative distribution function, CDF, for the Pareto distribution is defined by the following relationship:

CDF (x) = 1 – (xm/ x)k                                                                                                                                           (21)

for all x>xm, where k is the Pareto index. Then, the probability density function (PDF) is defined as:

PDF(x) = kxm(xm/x)k−1                                                                                                                                                (22)

Functional dependence of the probability density function on income allows for the exact calculation of total population in any income bin, total and average income in this bin, and the contribution of the bin to the corresponding Gini ratio because the PDF defines the Lorenz curve.

The actual estimates of index k reveal clear age dependence [Kitov, 2008a]. The evolution of the Pareto law index was estimated as the slope of linear regression line in the log-log scale. Using the CPS PIDs in various age groups aggregated over several years we obtained: k=3.91 for the age group between 25 and 34 years; k=3.48 between 35 and 44; k=3.38 between 45 and 54; k=3.14 in the age group between 55 and 64. It is clear that index k declines with age. Obviously, a smaller index k corresponds to an elevated PID density at higher incomes. The observed decrease in k with age deserves a special examination and should be inherently linked to some age-dependent dynamic processes above the Pareto threshold. The declining k is a specific feature of the age-dependent PIDs, which is not incorporated in our model yet.

For the entire population of 15 years of age and over Kitov [2008b] estimates k =3.35. It is close to the estimate in the age group between 45 and 54 years. This is not a coincidence since the number of people in the Pareto distribution is also a function of age and the potion of population with the highest incomes is the largest between 45 and 54 years of age, as discussed later in this Section. As a result, this age group has the largest input to the entire population in the Pareto range. Thus, the power law index for the entire population is practically the same as in this age group. For numerical calculations, we fix k=3.35 as estimated from the overall PIDs. The bias introduced by this choice into various income estimates for other age groups diminishes with their representation in the highest income range. As shown below, the portion of rich people in the youngest and eldest age groups is negligible.

One can also expect that the age-dependent and overall k undergo some changes over time. The overall index may vary because of the changing age pyramid and the time needed to reach the peak income Tc. In other words, the overall index may change because the input of various ages varies with time. Here, we study the gender-related difference in k, which can also be age-dependent. The observed income disparity between men and women may also be expressed in their presence among the richest share of U.S. population.

We have modelled the number of people above MP=0.43 from 1930 to 2011 [Kitov and Kitov, 2013]. Left panel in Figure 8 displays the predicted and observed numbers of people above the Pareto threshold in 1962 and 2011. We have measured these numbers from the annual PIDs borrowed from the IPUMS. Here, we have to stress that we used the entire working age population as the model input and calculated the whole period between 1930 and 2011 using only real GDP per capita as defining parameters. All other constants and initial values were fixed in 1930 and their evolution was defined by GDP growth only. The microeconomic model covers more than 80 years and gives correct predictions for two randomly selected points.

The fit between the measured and predicted numbers is excellent in various aspects. First of all, two curves for 2011 are close through the entire age range, except may be the youngest ages. The theoretical curve starts from 20 years and the observed one - from 18 years of age, but the latter curve is close to zero anyway. The measured 1962 curve is slightly higher than the predicted above the peak age. Overall, the model accurately predicts the age-dependent number of people with the highest incomes in two different years. At the same time, the predictions for 1962 and 2011 are coherent in terms they are calculated in one run with the same defining parameters and exogenous parameters (GDP and age pyramid) borrowed from official sources. This means that the model accurately predicts the evolution of each and every individual income and the Pareto threshold altogether.  Taking into account the successful prediction of the past values, one may use our model for projection of income distribution in the future. The microeconomic model describes all important aspects of income dynamics.

Figure 8. Left panel: The measured and predicted number of people with income above the thresholds $7,000 in 1962 and $73,000 in 2011. Right panel: The curves in the left panel are normalized to their respective peaks. The age of peak portion shifts from 41 years to 51 years.

All curves in the left panel of Figure 8 have clear peaks and then the number of people falls to zero at the age above 75; no elder people can be found in the Pareto income zone. In order to highlight the relative dynamics above the Pareto threshold we have calculated the portion of people above MP for all ages and then normalized the obtained portion curves to their peak values. In the right panel of Figure 8 we present the normalized portion of people who has reached the Pareto threshold as a function of age. This is the best illustration of the change in Tc, at least the peaks are sharper than in the mean income curves (see Figure 1). The latter contain two ingredients – low-middle (sub-critical) incomes and higher (super-critical) incomes. From Figure 8, we estimate Tc=27 years in 1962 and Tc=38 years in 2011. The difference between 1962 and 2011 is 11 years. Considering the accuracy of measurements, these estimates are in a good agreement with those obtained in Section 1.5 for Tc as a function of real GDP per capita. Such a big change has not been discussed in income-related economic literature yet.

There is an ongoing problem with the accuracy of the highest income measurements associated with confidentiality. Since the population with top incomes is represented in the CPS universe by a few people their actual incomes are “topcoded”, i.e. reduced to income bin boundaries [e.g., Larrimore et al., 2008]. According to IPUMS [2015], topcoding is defined as “a determination by the CPS that some high values were too sparse and specific to be recorded as they were reported to the CPS without the possibility of identifying the respondents.” The bias introduced by topcoding into the mean income estimates is not the only problem. It is very unfortunate for quantitative analysis that the topcodes are prone to severe revisions by income sources and by year. Moreover, the personal income estimates above the topcode were processed in different ways over time, i.e. they were changed according to different rules. In any case, all these procedures result in lower incomes reported by the CPS than those in reality. The artificial difficulties related to the topcoding deserve detailed investigation [e.g., Burkhauser et al., 2011].

The richest people make a significant share of the total personal income and the topcoding may introduce a measurable bias in some aggregate estimates like average income. For our model, the distortion of top incomes is not relevant, however. First of all, there are age groups where the effect of topcoding is marginal. For the youngest people, the portion of people in the Pareto distribution range is negligible while the dynamics of income growth at the initial segment of work experience is the most prominent. Figure 1 demonstrates that young people raise their incomes from zero to 60% of the peak mean income in the first five to ten years. As one can see in Figure 8, the observed portion of rich population in 2011 is less than 1% for ages between 15 and 25 and then starts to grow at a high rate. The curves in Figure 5 evidence that the mean income of the 22-year-olds is 30% of the peak mean income measured for the 50-year-olds. As a benefit of real economic growth for quantitative modelling, the period needed to enter the top income range increases with real GDP per capita, and thus, the effect of topcoding starts at larger work experience. By good fortune, the key parameters of our model, the dissipation factor, α, and the minimum size of work instrument, Λmin(τ0), can be most accurately estimated using the initial segment of the growth trajectory.
Secondly, the deviation from a power law distribution and errors in income estimates related to the CPS income topcoding do not affect the portion of people above the Pareto threshold. As we discussed in the beginning of this Section, there is no physical link between these two processes in the long-term observations and in the model. Effectively, whatever process disturbs the distribution of top incomes it is not driven by and does not drive the processes of income growth below the Pareto threshold. There exists only one connection between the people in the low/middle income range and the rich with the top incomes – the portion of people above the Pareto threshold. Figure 8 proves that our model exactly predicts this portion for the entire period with measurements.
The CPS observations show that the processes controlling the top incomes and those in the low-middle income are not linked. The rich and not-rich are not competing for the same personal incomes, at least for incomes from the sources included in the CPS questionnaire. The causes of the accelerated income growth in the top percentiles are in the focus of political, social as well as economic [Atkinson and Piketty, 2007; Atkinson et al., 2009; Burkhauser et al., 2012] discussions. They are beyond the scope of our model since the measures of income inequality are likely biased or/and misinterpreted in these discussions and they are related to the change in formal assignment of income sources to personal incomes rather than to real economic processes [Kitov, 2014].
Thirdly, the age of peak mean income, Tc, does not depend on the absolute value of the portion of people in the Pareto zone and the exponent, k, of the corresponding power law. It is defined by the sub-critical processes only. As Figure 1 proves, the peak age is the same for the CPS and IRS. This is because the sources of top incomes do not eat money from the sources of low-middle incomes.
Our model includes all necessary parameters to describe the distribution of top incomes, whatever are their sources and changes over time. We use the CPS estimates because they provide the most consistent and longer time series. As discussed above, the input of top incomes can vary with time, but such variations are fully accounted for by the changes in CPS estimates. For a quantitative model, the measured portion of true personal income should be constant over time. The CPS data are the closest to this requirement.


The income critical age

The exponential growth trajectory of income described by equation (4) clearly does not present the full picture of income evolution with age. As numerous empirical observations show (e.g., Figure 1), the average income reaches its peak at some age and then starts declining. This is seen in individual income paths, for instance presented in Mincer [1974]. In our model, the effect of exponential fall is naturally achieved by setting the money earning capability Σ(t) to zero at some critical work experience, t=Tc.

The solution of (4) for t>Tc then becomes:

ij (t) = ij (Tc) exp[−(1/Λmin)( γ̃/j)(t Tc)]                                                                    (16)

and by substituting (12) we can write the following decaying income trajectories for t>Tc :
ij(t) = ΣminΛminij{1 − exp(−(1/Λmin)(α̃/j)Tc)]exp{−(1/Λmin) (γ̃/j)(t Tc)}         (17)

First term in (17) is the level of income rate attained at Tc. Second term expresses the observed exponential decay of the income rate for work experience above Tc. The exponent index γ̃ represents the rate of income decay that varies in time and is different from α̃. It was shown in Kitov [2005a] (and also seen in Figure 1) that the exponential decay of personal  income rate above Tc results in the same relative level at the same age, when normalized to the maximum income for this calendar year. This means that the decay exponent can be obtained according to the following relationship:

γ̃ = −lnA / (TA Tc)                                                                                                   (18)

where A is the constant relative level of income rate at age TA. Thus, when the current age reaches A, the maximum possible income rate ij (for i = 29 and j = 29) drops to A. Income rates for other values of i and j are defined by (17). For the period between 1994 and 2002, the empirical estimates of parameters in (18) are A=0.45 and TA=64 years (see [Kitov, 2005a] for details).

The critical age in (16-17) is not constant.  For example, Figure 1 demonstrates that Tc has been increasing between 1962 and 2011. Therefore, its dependence on the driving force of income distribution - real GDP per capita - has to be one of central elements of our model since any model should match the long-term observations. To predict the increase in Tc(τ) we use (14): the time needed to reach some constant income level is proportional to the square root of real GDP per capita. Assuming that the peak value of the mean income is constant in relative terms, we obtain:

Tc(τ) = Tc(τ0) [Y (τ) / Y (τ0) ]1/2                                                                                   (19)

Figure 6 illustrates the growth in critical work experience, Tc, since 1930. The curve in the left panel illustrates the time dependence and is best interpolated by a straight line with a slope of 0.28 years per year as if the real GDP per capita grows as t2. During the last recession, the critical age dropped from 40.3 years in 2007 to 39.3 years in 2009. In the right panel, the dependence on GDP is shown for the same period.

Figure 6. Left panel: Secular increase in Tc is driven by the growing GDP per head. Right panel: The evolution of Tc as a function of GDP growth.

Above Tc, people can only use their earning instrument, which is growing with time, but their capability remains at zero level and income experiences an exponential decay. Formally, the size of work instrument cannot be zero since the dissipation term would be infinite. But we can easily imagine zero capability to earn money as the absence of interest to work. The model attributes positive capability to everyone in the working age population before Tc. This means that each and every person in a given economy must have a nonzero income. This is not what the CPS reports – approximately 10% of the working age population reports no income from the sources included in the CPS questionnaire.

When predicting incomes, we use the entire population. When comparing with observations, we include the zero-income CPS population into the income bin starting with $0 and recalculate the whole statistics like average income, the portion of people above a given threshold, etc. According to strict guidelines adapted in physics one should not calculate any aggregated characteristics of a closed system using only part of it. Such estimates are always biased and subject to fluctuations.

As an alternative to formal introduction of zero capability, one could claim that there exists a strong external process, which forces the exponential fall on top of the grown related to the original capacity to earn money. This does not resolve the problem, however, since description and explanation of these forces is needed. In addition to the homogeneous coverage of all population these forces should include the change in start time, i.e. should explain the growth in the age of peak mean income. We do not know any candidate.

Initial exponential growth and following decay, however, do not complete the model. Figure 2 shows, that our equation for income growth is not able to predict a power law distribution.  We still need to introduce special treatment for the very top incomes that have been shown to follow the Pareto distribution in multiple empirical studies.