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 smallestS 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 andL29 to reach the maximum income rate (see Figure 4), while persons with S1and 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(?). 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(?0)=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(?0)=0.43 for any starting year, where Y(?0)=1  Then the Pareto threshold evolves with time proportionally to growth in real output per capita:

MP (?) = MP(?0) [(?) / (?0)]  = MP(?0(?)                                                                       (20)