Here, a microeconomic model is presented, which has been developed to quantitatively describe the dynamics of personal income growth and distribution [Kitov, 2005a]. The model is based on one principal assumption that each and every individual above fifteen years of age has a personal capability to work. In essence, the capability to work is equivalent to the capability to earn money. To get money income, individuals have to use one or several means or tools from the full set of options that may include paid job, government transfers, bank interest, capital gain, inter-family transfers, and others. The U.S. Census Bureau questionnaire  lists tens of money income components. It is important to stress that some principle sources of income are not included in the CB definition, which results in the observed discrepancy between aggregate (gross) personal income (GPI), as reported by the Bureau of Economic Analysis and the gross money income calculated by the CB.
In this section, we summarize the formulation of a theoretical model, originally described in Kitov [2005a], and present it as a closed-form solution in a simplified setting. Figure 1 illustrates a few general features any consistent model has to describe quantitatively. In the left panel, we display the evolution of mean income curves from 1962 to 2013. The original income data are borrowed from the Integrated Public Use Microdata Series (IPUMS) preparing and distributing data for the broader research community [King et al., 2010]. These are income microdata, i.e. each and every person from the IPUMS tables is characterized (among other features) by age, gender, race, gross income, and the population weight, which allows projection of the individuals from the CPS population universe to the entire population. Using age, income, and population weight we have calculated the age-dependent mean income for all years and then normalized them to their respective peak values. The normalized curves better illustrate the growth in the age of peak income – from below 40 in the earlier 1960s to 55 in the 2010s. This is a sizeable change likely expressing the work of inherent mechanisms driving the evolution of personal income distribution. One cannot neglect the effect of increasing age when people reach their peak incomes – neither from a theoretical nor from the practical point of view.
In the right panel of Figure 1, we compare various mean income curves reported by two different organizations responsible for income measurements: the Census Bureau (CPS) and the Internal Revenue Service (IRS). The latter organization does not publish the age distribution of income on a regular basis and only the year of 1998 is available for such a comparison. The IRS mean income is calculated in 5-year age cells [IRS, 2015], the CPS prepares historical datasets with a 5-year granularity since 1993, and the annual estimates are available from the IPUMS microdata. The annual curve has also been smoothed with a nine-year moving average, MA(9). As in the left panel, all curves are normalized to their peak values.
There are significant differences in income sources and population coverage used by the CPS and IRS [Kitov, 2014]. Nevertheless, between 40 and 60 years of age, all curves in the right panel of Figure 1 are close to each other. With regard to the age of peak income, the CPS and IRS give identical results to the extent the age aggregation allows. The IPUMS curve has been smoothed and thus might have a slightly biased peak age. Between 25 and 40 years of age, the difference in normalized mean income is larger - likely because of the difference in income sources. The same effect is observed in the eldest age groups, where taxable incomes are not so often and the CPS curve is above the IRS one.
The closeness of the peak ages measured by the IRS and CPS is important for model applicability and reliability. The accuracy of income measurements, the coverage of population and income source, the level of historical consistency in income definition and survey methodology, the entire diversity of personal characteristics, and the length of time series provided by the Census Bureau all these features make it inevitable to use the CPS data for quantitative modelling. The reverse side of this choice is the necessity to defend the modelling results against the accusation that the CPS data are not full and representative.
It is true that the CPS misses some important sources of higher incomes, but Figure 1 stresses that the estimates of key features are not different if the IRS sources are included and some CPS income sources are excluded [Henry and Day, 2015, Ruser et al., 2004; Weinberg, 2004; U.S. Census Bureau, 2015b]. Besides, the CB provides the best income estimates for the poorest population, where incomes are just several dollars per year. Other organizations ignore small incomes. As a result, the estimates of personal income inequality based on the IRS data exclude half of the population, the poorest half. It is difficult to consider such estimates as accurate and helpful for understanding the mechanisms of the income distribution. The BEA income data are worthless for quantitative analysis of individual incomes - no age, gender, race information is available.
Astoundingly, the principal features observed in Figure 1 can be accurately approximated by basic mathematical functions. Moreover, these functions represent solutions of simple ordinary differential equations. The solid red line in the right panel is calculated to fit the CPS mean income curve. For this line, the equation is [1 - exp(-0.071(t-18))] + 0.09, where t is the age. The overall fit between the measured and approximating curves is extremely good from 18 to 55 years of age before the mean income curve starts to fall.
The approximating equation is a well-known function often called the “exponential saturation function”. This function represents a closed-form solution of a simple ordinary differential equation dx(t)/dt=a-bx(t), where a>0 and b>0 are constants. The match between the observed and approximating curves provides some hint on the forces behind income growth. The second term in the above equation represents the force counteracting the unlimited growth of x(t). The amplitude of the counteracting force is proportional to the attained level, and that implies the finite value of x(t)<Xmax, t →∞.
A standard example in general physics to illustrate the saturation process is associated with the heating of a metal ball by an internal source with constant power, U. The growth in temperature, T, is balanced by energy loss through the surface, and the energy flux through the surface is proportional to the attained temperature. Thermal conductivity can be treated as infinite in terms of the characteristic time of all other processes. For a ball of radius R and volumetric heat capacity, Cv, one can write the following equation:
4/3πR3CvdT(t)/dt = U – DT(t)4πR2 (1)
where D is a constant defining the efficiency of heat loss through the surface, which is similar to dissipation. By dividing both sides of (1) by 4/3πR3Cv we obtain:
dT(t)/dt = Ũ – D̃T(t)/R (2)
where Ũ=3U/(Cv4πR3) is the specific power of the heating sources expressed in units of thermal capacity, and D̃ = 3D/Cv. The solution of (2) is as follows:
T(t) = T0 + (ŨR/D̃)[1 - exp(-D̃t/R)] (3)
Relationship (3) implies that temperature approaches its maximum value ŨR/D̃ along the saturation trajectory, which we also observe in Figure 1. Instructively, the maximum possible temperature is proportional to R. This fact is helpful and important for a better understanding of our model and income observations. We interpret temperature as income, which one can reach using some physical capital, say, 4/3πR3, and personal efforts, say, U. Then the saturation curve in Figure 1 becomes an obvious result.
Above the age of peak mean income in Figure 1, one observes an exponential fall. The Blue dotted line is defined by function exp[-0.052(t-56)]. It best matches the IRS curve above 56 years of age. The match between the observed curve and the exponent is extraordinary even in terms of the hard sciences. The exponential function is a solution of a familiar equation: dx(t)/dt=-bx(t). The only difference is in the absence of term a, but now the curve starts from 1.0. The evolution of mean income measured by the IRS above the critical age can be expressed by a differential equation formally identical to that describing free cooling of a preheated sphere, i.e. when heating source U=0 in (1).
Hence, the observed features of the mean income behaviour are similar to those observed in simple physical experiments. However, we need to describe the income trajectory for each and every person in a given economy. It is natural to suggest that all individual incomes follow their own saturation curves and their average value follows up some individual trajectory. Then the distribution of parameters defining individual trajectories, i.e. income analogues of R and U, is completely constrained by observations. This is the intuition behind our microeconomic model.
Originally, the idea of income modelling with equation (2) came from geomechanics [Rodionov et al., 1982]. An identical equation describes the growth of stress, σ(t), in an inhomogeneous inclusion with characteristic size L experiencing deformation at a constant rate ε̇ as induced by external forces. Solution (3) is important to predict the highest possible level of stress at a given inclusion with size L. Unlike in the simple experiment with the heated sphere of radius R, the sizes of inhomogeneous inclusions are distributed according to a power-law L3dn/d(lnL) =const, where n is the number of inclusion of size L in a unit volume. This distribution defines the structural self-similarity of fractals.
Let us consider that deformation starts at time t0 and all stresses are zero before. Then stresses will rise at different rates for different inclusion sizes. At time t, there is some inclusion with size LM, which reaches its highest possible stress balancing deformation and dissipation. At all bigger inclusions, stress is still growing. When the rate of deformation is high enough and there are big enough inclusions the attained stress may exceed at some point the critical stress of fracturing. Then a quake may occur. This is a transition to a super-critical regime and the sizes of earthquakes are distributed by a power law.
In economics, higher incomes are characterized by a similar distribution, but they are the net result of all forces and agents in the economy, which both vary with time. They do not represent a predefined structure as in geomechanics. Moreover, low and middle incomes are distributed according to an exponential law rather than a power one. So, we had to construct the basic distributions of defining parameters, which result in exponential distribution of low-middle incomes and power-law distribution above the Pareto threshold. The process of model development with explicit differential equations together with the selection of underlying distributions is described in the following Subsections.