Microeconomic model for income distribution

1.1.Ordinary differential equation of personal income growth

On the whole, two main driving forces of our model are similar to those in the Cobb-Douglas production function: Y=W^aK^b, where Y is the measure of production (e.g., Gross Domestic Product) in a given country, which may be measured in the country-specific currency, W is the labour often considered as work hours, K is the physical (or work) capital (e.g., machinery, equipment, buildings, hardware, software, etc.), and a and b are the output elasticities.  Indeed, labour is the only source of products measured in money, and thus, the only source of income. At the same time, using larger and more efficient work instruments people produce more goods and services, also in terms of their real value measured in money units. This consideration is fully applicable at the level of individual production. All persons of working age are characterized by nonzero (and varying) capabilities to generate income and use work instruments of different sizes to do that.

Unfortunately for economics, the Cobb-Douglas function is a non-physical one. It implies the unlimited growth in GDP because it does not include any forces counteracting the production. Following the physical approach discussed in Section 1.1, we assume that no person is isolated from the surrounding world. When a person starts her work the forces arise to counteract any production action. In this setting, the work (money) she produces must dissipate (devaluate) through the entire diversity of interactions with the outside world, thereby decreasing the final income per unit time. All counteractions with outer agents, which might be people or some externalities, determine the final price of the goods and services the person produces.

Following the shape of mean income curve in Figure 1, the evolution of personal income has to be described by a phase of quasi-linear growth in the initial stage of work experience, by a saturation function during the prime working age, and an exponential decline following the peak income. Given the differences between individuals, these three stages may develop at different rates. In Section 1.1, we have discusses similar trajectories and found that a larger body undergoes faster heating because it loses relatively less energy and also reaches a higher equilibrium temperature.

To characterize the change in individual income we introduce a new variable - income rate, M(t), the total income person earns per year. For the sake of brevity we further call M(t) “income”. In essence, M(t) is an equivalent of Y in the Cobb-Douglas production function. The principal driving force of income growth is the personal capability to earn money, σ(t), which is an equivalent of labour, W, in the Cobb-Douglas function. The meaning of the capability to earn money differs from that usually implied by the notation “human capital”. Obviously, the level of human capital of many distinguished scientists is extremely high while their capability to earn money might be extremely low.

Applying our physical intuition to income, we assume that the rate of dissipation of income has to be proportional to the attained level of M(t). The equation defining the change in M(t) should include a term, which is inversely proportional to the size of means or instruments used to earn money, as defined by variable Λ(t). Then the dissipation term is proportional to M(t)/Λ(t). Following the analogy in Section 1.1, one can write an ordinary differential equation for the dynamics of income depending on the work experience, t:

dM(t) / dt = σ(t)  − αM(t) / Λ(t)                                                                                  (4)

where M(t) is the rate of money income denominated in dollars per year [$/y], t is the work experience expressed in years [y]; σ(t) is the capability to earn money, which is a permanent feature of an individual [$/y²]; Λ(t) is the size of the earning means, which is a permanent income source of an individual [$/y]; and α is the dissipation factor [$/y²].

We assume that σ(t) and Λ(t) are mutually independent - that is a person’s ability is unrelated to her work instrument. Notice that we have chosen t to denote the work experience rather than the person’s age. It is natural to assume that all people start with a zero income, M(0)=0, which is the initial condition for (4). At the initial point, t = 0, when the person reaches the working age (15 years old in the USA) her income is zero and then changes according to (4) as t>0. Note that both σ(t) and Λ(t) can vary with t. This means that (4) has to be solved numerically, which is the approach we apply to calibrate the model to data. Before proceeding to the calibration stage, we first make a few simplifying assumptions, under which the model has a closed-form solution.

For the sake of simplicity, which will be explained later on, we introduce a modified capability to earn money:

Σ(t) = σ(t)/α                                                                                                                (5)

From this point onwards we will omit the word "modified" and refer to Σ(t) simply as earning capability or ability. For the completeness of the model, we introduce second time flow, τ, which represents calendar years. The time flow for work experience, t, and calendar years, τ, relate to each other in a natural fashion. For a simple illustration, consider a person that turns 15 in a year τ₀, i.e. her work experience is t₀ = 0. By year τ this person will have t =τ − τ₀ years of work experience. Consequently, τ is a global parameter that applies to everyone, whereas t is an individual characteristic and changes from person to person. We allow Λ and Σ to also depend on τ, thereby introducing differences in income capability and instrument among age cohorts. In other words, the model captures cross sectional and intertemporal variation in both parameters. In line with the Cobb-Douglas production function, we make a simplifying assumption by letting Λ(τ₀,t) and Σ(τ₀,t) to evolve as the square root of the increment in the aggregate output per capita. The capability and instrument thus evolve according to:

Σ(τ₀,t)  = Σ(τ₀,t₀) [Y (τ) / Y (τ₀) ]^1/2                                                                               (6)

Λ(τ₀,t)  = Λ(τ₀ t₀) [Y (τ) / Y (τ₀) ]^1/2                                                                               (7)

where Σ(τ₀,t₀) and Λ(τ₀,t₀) are the initial values of capability and instrument for a person  with zero work experience in year τ₀; Y(τ₀) and Y(τ) are the aggregate output per capita values in the years τ₀ and τ, respectively, and dY(τ₀,t)=Y(τ)/Y(τ₀)=Y (τ₀+t)/Y(τ₀) is the cumulative output growth. Note that the initial values Σ(τ₀,t₀) and Λ(τ₀,t₀) depend only on the year when the person turns 15, τ₀, since the initial work experience is fixed at t=0 for all individuals irrespective of when they start working. Now we can restrict our attention to the initial values of the capability and instrument as functions of the initial year: Λ(τ₀) and Σ(τ₀), respectively. The product of equations (6) and (7), Σ(τ₀,t₀)Λ(τ₀,t₀), evolves with time in line with growth of real GDP per capita as in the Cobb-Douglas production function. We call ΣΛ the capacity to earn money, which means that Λ(τ₀,t₀)Σ(τ₀,t₀) is the initial capacity.

Equation (4) can be re-written to account for the dependence on the initial year, τ₀:

dM(τ₀,t) / dt = α{Σ(τ₀,t) − M(τ₀,t) / Λ(τ₀,t)}                                                                 (8)

Note that when we fix τ₀ and restrict our attention to a person with work experience t, we return to our original equation (4). Moreover, the path of income dynamics depends on τ₀ only through the influence of the latter on the initial earning capability and instrument. In other words, τ₀ only determines the starting position of the income rate and not the trajectory of the income path, which is completely described by equation (4).

1.2. Distribution of capability and instrument size

Actual personal incomes in any economy have lower and upper limits. It is natural to assume that the capability to earn money, Σ(τ₀,t), and the size of earning means, Λ(τ₀,t), are also bounded above and below. Then they have positive minimum values among all persons, k = 1, . . . ,N, with the same work experience t in a given year τ₀: minΣ_k(τ₀,t)=Σ_min(τ₀,t) and minΛ_k(τ₀,t)=Λ_min(τ₀,t), respectively,  where Σ_k(τ₀,t) and Λ_k(τ₀,t) are the parameters corresponding to a given individual. We can formally introduce the relative and dimensionless values of the defining variables in the following way:

S_k(τ₀,t)  = Σ_k(τ₀,t) / Σ_min(τ₀,t)                                                                                         (9)

and

L_k(τ₀,t) = Λ_k(τ₀, t) / Λ_min(τ₀,t)                                                                                      (10)

where S_k(τ₀,t) and L_k(τ₀,t) are the dimensionless capability and size of work instrument, respectively, for the person k, which are measured in units of their minimum values. So far, all N persons in the economy are different and at this stage of model development we need to introduce proper distributions of S_k(τ,t) and L_k(τ,t) over population as well as their functional dependences on time and age.

The complete description of the development of discrete uniform distributions for S_k and L_k by matching predicted and observed distributions of personal income in the U.S. is presented in [Kitov, 2005b]. Here, we use the final outcome. Specifically, the relative initial values of S_k(τ₀,t₀) and L_k(τ₀,t₀), for any τ₀ and t₀, have only discrete values from a sequence of integer numbers ranging from 2 to 30. For any k, there are 29 different values of S_i(τ₀,t₀) and L_j(τ₀,t₀): S₁(τ₀,t₀)=2, . . . , S₂₉(τ₀,t₀)=30, and similarly for L_j(τ₀,t₀), where j=1,...,29. Assuming uniform distribution between 29 different capabilities, we get that the entire working age population is divided into 29 equal groups. All k work instruments are uniformly distributed over 29 different sizes from 2 to 29.

The largest possible relative value S_max=S₂₉=L_max=L₂₉=30 is only 15 times larger than the smallest S_min=S₁= L_min=L₁=2. In the model, the minimum values Σ_min and Λ_min are found to be two times smaller than the smallest possible values of L₁ and S₁, respectively. Because the absolute values of variables Σ_i, Λ_j, Σ_min and Λ_min evolve with time according to the same law described in (6) and (7), the relative and dimensionless variables S_i(τ,t) and L_j(τ,t), i, j = 1, . . . , 29, do not change with time thereby retaining the discrete distribution of the relative values. This means that the distribution of the relative capability to earn money and the size of the earning means is fixed over calendar years and age cohorts. The rigid hierarchy of relative incomes is one of the main implications of the model and is supported empirically by the PIDs reported by the CB for the period between 1993 and 2011 [Kitov, 2005a,b; Kitov and Kitov, 2013]. The proposed uniform distributions are rather operational and should not be interpreted far beyond their capability to model actual distribution of personal income. For example, in this paper we lift strict assumptions of the original model in order to match the difference in income distribution between males and females. At the same time, the good fit between observations and predictions provide a solid basis to interpret observations in term of model parameters, as it adapted in physics.

The probability for a person to get an earning means of relative size L_j is constant over all 29 discrete values of the size and the same is valid for S_i. In a given year τ, all people are distributed uniformly among 29 groups of the relative ability and over 29 groups of instruments to earn money, respectively. The distribution over income involves the history of work experience t described by (4), and thus, differs from the distribution over relative values. The relative capacity for a person to earn money is distributed over the working age population as the product of the independently distributed S_i and L_j:

S_i(τ,t)L_j(τ,t)  = {2×2 ,...,2×30, 3×2 ,...,3×30,..., 30×30}

There are 29×29=841 different values of the normalized capacities available between 4 and 900. Some of these cases seem to be degenerate (for example, 2×30=3×20=4×15=...=30×2). However, Σ and Λ have different influence on income growth in (4) and each of 841 S_iL_j combinations define a unique time history.

It is worth noting that our model does not predetermine actual income trajectory for real people. The model assumes that real people have incomes, which can only be chosen from 841 individual paths predefined for their year of birth. (The exception is when personal incomes reach the Pareto threshold, as discussed in the following Sections. The Pareto distribution also fixes all individual incomes, however.) This statement is equivalent to the observation that the PIDs reported by the CPS are repeated year by year, i.e. the portion of people in a given range of total income share is rock solid, and thus, the observed Gini ratio is constant.

Figure 2. Left panel: The probability density function, PDF, of the personal capacity, pc=SL, distribution as defined by the independent uniform distribution of S_i and L_j. The PDF is well approximated by an exponential function 0.033exp(-2.9pc) between 0.08 and 0.8; then the PDF falls faster than the approximating exponent. Right panel: comparison of observed and predicted PDFs in 2001. The independent distribution of S and L fit the oscillations in the observed PID for people between 60 and 65 years of age.

Left panel of Figure 2 depicts the probability density function (PDF) for the distribution of the capacity to earn money, SL. The underlying frequency distribution was obtained in 0.01 bins of personal capacity. For the lowermost incomes, we observe a local minimum. After the PDF reaches its peak value, it falls as an exponential function 0.033exp(-2.9pc) between 0.08 and 0.8. In the range of the highest personal capacities, the PDF falls faster than the exponent approximating the mid-range values.  In the right panel of Figure 2, we illustrate the essence of the uniform and independent distributions of S and L. We have calculated a probability density function using the PID for people between 60 and 65 years of age as reported by the CPS in 2001. Equation (1) suggests that many people had to reach their maximum incomes, SL, at the age above 60, and thus, the PDF for the real PID has to fit the theoretical distribution in the left panel. The only difference in that we have recalculated the theoretical PDF in the personal capacity bins corresponding to actual income bins of $2,500. The choice of discrete values between 2 and 30 is dictated by the fit of the observed and predicted PDFs in Figure 2. The independent distribution of S and L best fits the oscillations in the observed PID for people between 60 and 65 years of age. Any change in the range and start values (2 to 30) of S_i and L_j destroys the observed coherence in the PDFs’ fall rate and well as the synchronization in frequency and amplitude.

Figure 3 displays the cumulative probability function, CDF, for the theoretical PDF in Figure 2. The CDF is helpful in estimation of the portion of people above any threshold. We cut top 10% of the personal capacities and found that the threshold is 0.62. For the top 1%, the threshold is 0.9. These estimates are important for the further discussions of the share of people in the Pareto distribution, which is quite different from the quasi-exponential distribution below the Pareto threshold. Our model does not include any definition of “poverty” as a measure of the lowermost incomes. The CDF provides a useful tool to introduce an operational definition of relative poverty threshold. According to the World Bank, the relative poverty threshold is 50% of the mean income in a given country. Theoretically, the mean personal capacity to earn money is 0.283. Then the poverty threshold is 0.14. In Figure 3, red line shows that 32% of people are below the poverty line as defined by the personal capacity to earn money.

Figure 3. The cumulative density function, CDF, illustrates the rapidly decreasing portion of people with personal capabilities above some threshold: only 10% of population has the capacity to earn money above 0.62.

According the U.S. Census Bureau, the official poverty threshold in the U.S. for one person (unrelated individual) was $12,071 in 2014 and the mean income for population with income $42,789. The relative poverty threshold is then 0.08 in terms of personal capacity. It gives approximately 20% of the total working age population below the poverty line. The official level of poverty is approximately 14% of population with income. If to include 10% of population without income into the poverty statistics we obtain approximately 20% of total population as well. So, the underlying distribution of the personal capacity to earn money does predict the portion with the highest incomes and the level of poverty.   

1.3. Numerical modelling, personal trajectories, early rise

Since the model contains time varying parameters, we use numerical methods to solve it and calibrate to data. However, in order to better understand the system behaviour we first consider a simplified case when Σ(τ₀,t) and Λ(τ₀,t) are constant over t. It is a plausible assumption since these two variables evolve very slowly with time. Note that in the following exposition we fix τ₀ and so income trajectories are a function of work experience t only. Now, given constant Σ and Λ, as well as the initial condition M(0)=0, the general solution of equation (4) is as follows:

M(t) = ΛΣ[1 – exp( - αt/Λ) ]                                                                                      (11)

Equation (11) indicates that personal income growth in the absence of economic growth, i.e. dΛ/dt=dΣ/dt=0, depends on work experience, the capability to earn money, the size of the means used to earn money.

It is possible to re-arrange equation (11) in order to construct dimensionless and relative measures of income. We first substitute in the product of the relative values S_i and L_j and the time dependent minimum values Σ_min and Λ_min for Σ and Λ. (For notational brevity we omit the dependence of parameters on time and experience.) We also normalize the equation to the maximum values Σ_max and Λ_max in a given calendar year, τ, for a given work experience, t. The normalized equation for the rate of income, M_ij(t), of a person with capability,  S_i, and the size of earning means, L_j , where i, j ={2, . . . , 30} is as follows:

M_ij(t) / [S_maxL_max] = Σ_minΛ_min(S_i/S_max)(L_j/L_max){1 – exp( – αt/[(Λ_minL_max)(L_j/L_max)]}   (12)

or compactly:

M̃̃_ij(t) = Σ_minΛ_minS̃_iL̃_j[1 – exp ( − t(1/Λ_min)(α̃/L̃_j))]                                                        (13)

where

M̃_ij(t) = M_ij(t)  / (S_maxL_max)

S̃_i = S_i / S_max

L̃_j = L_j / L_max

α̃ = α / L_max

and S_max=L_max=30. In this representation, S̃_i and L̃_j range from 2/30 to 1. The modified dimensionless dissipation factor α̃ has the same meaning as α in (4).

Note that Σ and Λ are treated as constants during a given calendar year, but evolve according to (6) and (7) as a function of time. The term Σ_min(τ₀,t)Λ_min(τ₀,t) then corresponds to the total (cumulative) growth of real GDP per capita from the start point of a personal work experience, τ₀ (t₀=0), and vary for different years of birth. This term might be considered as a coefficient defined for every single year of work experience because this is a predefined exogenous variable. Relationship (13) suggests that one can measure personal income in units of minimum earning capacity, Σ_min(τ₀,t)Λ_min(τ₀,t), for each particular starting year τ₀. Then equation (13) becomes dimensionless and the coefficient changes from Σ_min(τ₀,t₀)Λ_min(τ₀,t₀)=1 in line with real GDP per capita. Further, we present simulations of individual income trajectories under the assumption of constant parameters and compare them to the calibrated version, where the output growth is taken into account and all defining parameters are allowed to grow.

For constant L_j and S_i, one can derive from (13) the time needed to reach the absolute income level H, where H<1: i="">

 

t_H = Λ_jln[1-H)] / α                                                                                                     (14)

   

This equation is correct only for persons capable to reach H, i.e. when L_jS_i/S_maxL_max>H.  With all other terms in (14) being constant, the size of work instrument available for a person, Λ_j, defines the change in t_H. In the long-run, t_H increases proportionally to the square root of the real GDP per capita.

Figure 4 illustrates two channels of t_H dependence on Λ_j. We consider two different values of S_i=2 and 30 and same value of L_j=30 and compare personal incomes curves in 1930 and 2011. For constant Λ_j, the time needed to reach HS_jL_i for a given person does not depend S_i – the curves in the left and right panels are pair-wise identical in terms of shape. This means that the person with S_i=2 reaches, say, 50% of her maximum personal capacity Λ_jΣ_i exactly at the same time as the person with S_i=30 reaches 50% of her maximum income. At the same time, the person with S_i=2 never reaches H=0.5 - her income ceiling is 1/15. As we discussed earlier, only 10% can reach the level of 0.62, and only in case they would have infinite time.

The increase in Λ_j from 1930 to 2011 results in a much slower income growth.  Solid lines in Figure 4 represent the solutions for constant Λ and Σ, and dotted lines represent the numerical solution of (13) with real GDP per capita. In 1930, the person with S=L=30 reaches H=0.4 in 4 years of work experience and it takes 8 years in 2011.  One can see that the numerically integrated curves are below the simple theoretical prediction. The increasing Λ does affect the relative level a person can reach before the critical age discussed in Section 1.1.

Figure 4. Growth trajectories of persons with two different capabilities to earn money (S_i) 2 and 30 and identical L_j=30. The increase in Λ_j from 1930 to 2011 results in slower income growth.  Solid lines represent the solutions for constant Λ and Σ, and dotted lines represent the numerical solution of (13) with real GDP per capita.

For the starting segment of income growth, when t<<1 b="" term="" the="">α

t/Λ in (11) is also << 1. One can derive an approximate relationship for income growth by representing the exponential function as a Taylor series and retaining only two first terms. Then (11) can be re-written as:

     

M_ij(t) = Σ_iΛ_j αt/Λ_j = Σ_iαt                                                                                            (15)

i.e. the money income, M, for a given person is a linear function of time since Σ_i  and α are both constants. Using (5), one regains the original meaning of the personal capability to earn money: M_ij(t)=σ_it. Figure 5 illustrates the existence of a linear segment in 1962 and 2011 as well as the increase of its duration as a result of decreasing α/Λ, as the size of working instrument Λ grows proportionally to the square root of GDP per capita.

Figure 5. The evolution of normalized mean income at the initial stage. The change in growth rate with age is well predicted by the model for 1962 and 2011 as well as the change in the trajectories induced by economic growth during 50 years. At the initial stage of work experience, the input of the highest incomes is negligible – almost no people distributed by a power law.  Notice that better measurements in 2011 are accompanied by a higher accuracy of prediction. In 1962, the observed fluctuations are related by poor population coverage for the youngest cohorts.

physical intuition behind income distribution

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 includes paid job, government transfers, bank interest, capital gain, inter-family transfers, and others. The U.S. Census Bureau questionnaire [2006] includes 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 IPUMS [2015]. 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 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 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 at 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 bins, 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]. Between 40 and 60 years of age, all curves in 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. 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 our consideration of the 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 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 be different if the IRS sources are included. 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 income inequality based on the IRS data exclude half of population, the poorest half. It is difficult to consider such estimates as accurate and helpful for understanding the mechanisms of income distribution. The BEA income data are absolutely worthless for quantitative analysis - no age, gender, race information is available.

 

Figure 1. Left panel: The change in the shape of mean income dependence on age from 1962 to 2013 as measured by the Census Bureau in the March Supplements of the Current Population Survey. All curves are normalized to their respective peak values. Right panel: Comparison of mean income dependence on age as measured by the Census Bureau (CPS) and the Internal Revenue Service (IRS). The only year with data available from the IRS is 1998.

It is very important to stress that the features observed in Figure 1 can be approximates by simple mathematical functions. Moreover, these functions represent solutions of simple ordinary differential equations. 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-15))] + 0.09, where t is the age. The overall fit between the measured and approximating curves is extremely good between 18 and 55 years of age, before the mean income curve starts to fall.

The approximating equation is a well-known function often called “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. 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)max

, t →∞.

A standard example in general physics to illustrate the saturation process is associated with heating of a metal ball by an internal source with constant power, W. The growth in temperature, T, is balanced by energy loss through the ball 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, C_v, one can write the following equation: 

           

4/3πR³C_vdT(t)/dt =  W – 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 if (1) by 4/3πR³C_v we obtain:

dT(t)/dt =  W̃ – D̃T(t)/R                                                                                               (2)

where W̃=3W/(C_v4πR³) is the specific power of the heating sources expressed in units of thermal capacity, and D̃ = 3D/C_v. The solution of  (2) is as follows:

            T(t) = T₀ + (W̃R/D̃)[1 - exp(-D̃t/R)]                                                                           (3)

Relationship (3) implies that temperature approaches its maximum value W̃R/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 better understanding of our model. We interpret temperature as income, which one can reach using some physical capital, say 4/3πR³, and personal efforts, say W. 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. 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 hard sciences. The exponential function is a well-known 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. Hence, 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 W=0 in (1).

Hence, the observed features of the mean income behaviour are similar to those observed in a simple physical experiment. However, we need to describe income trajectory for each and every person in a given economy. It is natural to suggest that all individual incomes follow own saturation curves and their average value repeats some individual trajectory. Then the distribution of parameters defining individual trajectories, i.e. income analogues of R and W, is completely constrained by observations. This is the intuition behind our 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 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 a structural self-similarity of fractals.

In economics, higher incomes are characterized by a similar distribution, but they are the net result of all forces and agents in the economy not the 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 the exponential distribution of low/middle income 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 subsection.

Income distribution: cross country comparison - Canada vs. USA

Canada vs. USA

We finish our country analysis with Canada, which provides an extended set of income data. These data cover all persons who completed specific tax return forms for the year of reference. Therefore, the data are related to tax purposes only and some non-taxable income sources are missing from the dataset. In 2013, about 74.9% of Canadians (of all ages) filed tax returns. Most children do not file tax return as well some elder people. That may introduce age-specific bias in the estimates for children and pensioners. Unlike the CPS survey, all data are extracted from administrative files. The sample includes 100% of individuals who filed an individual tax return but not all population. Depending on age, from 89% to 96% of the population is covered by administrative records [8]. Overall, the population and income source coverage is good for the purposes of our research.

Figure 19 depicts the evolution of real mean income since 1976 [9]. Three years age shown for comparison – 1976, 1993 and 2011, i.e. with approximately a 17 year step. The real mean income estimates are given in 2011 constant Canadian dollars and calculated in 10-year bins except the two youngest age groups: under 20 years – the average age of 17 years is used, and between 20 and 24 years of age. The curve for 2012 presents real mean incomes only in 10-year bins; it was obtained from a different table provided by Statistics Canada.

Figure 19. The evolution of age-dependent real mean income (in 2011 constant Canadian dollars) in Canada from 1976 to 2012. All estimates are given in 10-year bins except the two youngest age groups (below 20, and between 20 and 24). The curve for 2012 presents mean incomes only in 10-year bins and was obtained from a different table provided by Statistics Canada. The highly unusual feature is the 1976 curve above the 1993 curve.

The first and highly unusual observation is that the 1976 curve is above the 1993 curve. The real GDP per capita curve in Figure 5 gives $18,138 (1990 US$) in 1992 and $14,902 in 1976. For comparison, the mean income for the whole population with income in Canada is presented in Figure 20. In fact, the average income in 1976 was $35,700 and only $33,300 in 1993. The contradiction between real GDP per capita and mean income is likely related to the difference between domestic currency and that converted at Geary Khamis PPPs. In our quantitative analysis, this discrepancy may introduce large errors in theoretical estimates of the peak age.

Following the procedure developed in two previous paragraphs, we normalize the curves in Figure 19 to their respective peak value and present the central segments of the obtained curves in Figure 21. The 1976 curve peaks between 41 and 42 years of age (27 to 28 years of work experience). In 1992, the peak age is about 45 years, and then it reaches 49 to 50 years in 2012. Overall, the increase in the age of peak mean income increases by about 8 years. Theoretically, the change in real GDP per capita from $14,902 to $25,629 should change by a factor of 1.31, i.e. the peak working age experience rises from 27.5 years to 36.1 years (50 years of age). This is an extremely accurate prediction, especially taking into account the problems with real GDP per capita estimates. The proportion of working age population does not change much in the USA since the mid-70s. It is likely not a big error if we assume that the share of working age population in three studied countries does not change since 1976. Then the correction for population does not affect the relative change in real GDP per capita and the estimates of the peak age increase in Canada, New Zealand and the UK are not biased.

Figure 20. The evolution of mean income (2011 CAD) in Canada between 1976 and 2011.

Figure 22 is similar to Figures 10 and 16 and displays the evolution of peak-normalized mean income in various age groups. The peak age resides in the bin from 35 to 44 years of age before 1990. Then the peak jumps into the elder group and will likely stay below 54 years before the group between 55 and 64 overtakes the lead. As in other three countries, the proportion of mean income has been increasing in all elder groups and decreasing in the younger groups. In Canada, fluctuation of the curves in Figure 22 is much lower than that for New Zealand and likely at the same level as in the UK. This is a consequence of income data quality – better coverage of population is directly translated into the accuracy of mean income estimates.

Figure 21. Three curves in Figure 19 normalized to their respective peak values. The age of peak mean income value increases with time from 42 years in 1976 to 49 years in 2012.

Figure 22. The evolution of mean income in all 10-year age groups normalized to the peak value in the same year. In the younger age groups (“18”, “22”, and ”30”) the proportion of mean income has been falling since 1976.

Figure 23 depicts two panels where the normalized mean income curves observed Canada in 2011 ($22,994) and 1980 ($12,931) are matched by U.S. curves for 1987 ($21,788) and 1962 ($11,904), respectively. The excellent match between Canada2011 and USA1987 curves well corresponds to GDP estimates, while the GDP levels for 1980 in Canada and 1960 in the USA differ more in relative terms. As we discussed above, the GDP values for the USA before 1975 have to be corrected for the effect of changing working age population. Between 1987 and 1962 the correction is approximately 7%. This makes the 1962 estimate to increase to $12,708. So, Canada provides the longest time series of mean income among three studied countries with the largest factor of peak age increase - 1.31. This makes our measurements more precise and allows better fir between observed and predicted change in the peak mean income.

 

Figure 23. The evolution of mean income in Canada in 2011 and 1980 is matched by the curves measured in the USA in 1987 and 1962, respectively. Age bins are 10 years for Canada. For the USA, we use microeconomic data with annual mean income smoothed with a MA(9).

Canada is the only country from the studied trio reporting age-dependent PIDs for the higher incomes. Figure 24 displays the number of people with income above a given income threshold as a function of age. We have selected different thresholds to retain the portion of population: $100,000 in 2000, $100,000 in 2006, and $150,000 in 2013. All age bins are 10 years, except the youngest between 0 and 24 years of age. The youngest bin is prone to strong bias because it includes children with incomes. The 2006 curve is higher than the 2000 curve because they have the same threshold but the total nominal income in 2006 is much larger than in 2000, and thus, more people have larger incomes. Moreover, the population pyramid changing with time may introduce a significant bias into the number of people of a given age. In Figure 24, three curves peak at different ages. To suppress the population effect we have scaled the portion of people with the highest income to the total population with income in the same age bin. Figure 25 shows three normalized curves. As one can see, the age pyramid effect is removed and all curves peak at the same age. Summing the number of people above the threshold in all age groups and dividing it by the total population with income, we calculate the total portion of people with incomes above the threshold. For the thresholds in Figure 24, this portion is 2.5% in 2000, 4.5% in 2006, and 2.8% in 2013.

Figure 24. The number of people with income above a given income threshold as a function of age. Thresholds are $100,000 in 2000, $100,000 in 2006 and $150,000 in 2013. Age bins are 10 years, except the youngest between 0 and 24 years of age. The number of people depends on threshold and population in each bin.

Figure 25. The portion of people with income above a given income threshold as a function of age: the number of people above the threshold in a given age bin is divided by the total number of people in the same bin. The age pyramid effect is suppressed. The cumulative portion of people above the threshold is 2.5% in 2000, 4.5% in 2006, and 2.8% in 2013. Notice the same bin and threshold settings as in Figure 24.

When total income and population are subject to significant changes with time the best way to compare age-dependent income distributions in different years is to normalize them to their respective peak values. Then direct comparison is possible which shows the relative rate of income/population change with age. Figure 26 depicts three peak-normalized curves from Figure 25. These normalized curves practically coincide with just small differences in the age groups 25 to 34 years and 55 to 64 years. The age of the largest portion of people with the highest incomes resides in the bin between 45 and 54 years. Because of the width of this bin, which includes the true peak year for all curves, it is difficult to resolve any change.

Figure 26. The curves in Figure 25 are normalized to their respective peak values. The age of largest portion of people with the highest incomes resides in the bin between 45 and 54 years. It is difficult to estimate the change in peak age. Notice the same bin and threshold settings as in Figure 24.

One problem in Figure 26 is that the 2006 curve lies above the 2000 curve, while our model and experience suggest the opposite situation. This controversy is actually related to the difference in the total portion of people with the highest incomes: 2.5% in 2000 and 4.5% in 2006. All PIDs available for Canada between 2000 and 2013 are characterized by $50,000 income bins above $100,000. The choice of threshold is limited to the boundaries of these bins. The year of 2000 is characterized by the lowermost number of people and gross nominal income, and thus, by the lowermost threshold of the Pareto distribution. We use it to illustrate the change in the age-dependent portion of people with the highest incomes for several thresholds between $50,000 and $250,000. Figure 27 illustrates the change in the age-dependent curve. The peak age grows with the threshold, i.e. more and more time is needed to reach higher incomes. This observation puts an important constraint on the direct cross comparison of the age-dependent portion of people with the highest incomes. One should use thresholds, which retain the total portion of people at the same level.

In Figure 28 we match the 2013 Canada curve and the 1995 US curve. The coincidence between the portions of total population with incomes above $150,000 (CAD) in Canada and $82,500 (USD) in the USA, both in current dollars, is striking. We retain the total portion of population above these thresholds at approximately 2.7%. That eliminates the threshold dependent bias. Total Economy Database reports $26,000 (1990 US$) for Canada in 2013 and $24,712 for the USA in 1995. The difference is not large and a 4% correction for the change in working age population makes the 1995 U.S. estimate to rise to $25,643. Hence, the fit between two curves proves that the portion of people with the highest incomes is likely a country-independent variable. In other words, it is likely a universal variable which depends only of real GDP per capita. It is important to extend the set of countries in order to support this finding.

Figure 27. Canada 2000. The age-dependent portion of people with incomes above five thresholds - from $50,000 to $250,000. The peak age depends of threshold. Therefore, one has to compare curves with thresholds giving the same portion of people.

Figure 28. Comparison of the age-dependent portion of people with incomes above given threshold. The 2013 Canada curve (>$150,000, current CAD) is best fit by the 1995 US curve (>$82,500, current US$). TED reports for Canada $26,000 (1990 US$) in 2013 and $24,712 for the USA in 1995. The total portion of population above the thresholds in both cases is approximately 2.7%.

Conclusion

We have studied two specific features of personal income distribution in three countries: Canada, New Zealand, and the UK, and compared them with the USA. The dependence of mean income and the portion of people with the highest incomes on age are both characterized by varying length of the involved time series and their accuracy. Canada provides a set of mean income data covering practically the whole population and the period since 1976, but the data on personal income evolution with age are limited to the period between 2000 and 2013. Statistics New Zealand also reports tax-related income as obtained from a survey covering a small portion of the whole population. Higher amplitude fluctuations, likely induced by underrepresentation of the highest incomes, limit the usefulness of these estimates for the purposed of our study. In addition, there is some controversy between real GDP and income estimates reported by New Zealand and the Conference Board. The UK provides the shortest but useful time series collected by HMRC.

All data corroborate our main assumption – the evolution of income distribution is universal, follows a unique trajectory, and depends only on real GDP per capita converted at PPP exchange rates. We have proved quantitatively that the dependence of mean income on age in Canada, New Zealand and the UK as well as the age-dependent portion of people with the highest incomes in Canada do reproduce similar dependencies observed in the USA, but many years before, when the level of real GDP per capita was the same. Since the U.S. outpaces three studied countries by several thousand dollars per head, the lag reaches 20 to 25 years. For example, the mean income dependence measured in the UK in 2012-2013 one-to-one repeats that observed in the USA in 1992. The age-dependent portion of rich people in Canada in 2013 reproduces that measured in the USA in 1995. The time dependence of the studied characteristics is just parametric, however.

We have proven that the growth of work experience corresponding to the peak mean income is accurately described by the square-root function of real GDP per capita. This feature corresponds to the key assumption of our model, which predicts both studied features precisely. The coherence of theoretical predictions and long-term observations in four countries proves that the evolution of personal income (at least in these four countries) is a physical process described by a simple relationship. 

The results of measurements carried out in this study well match the prediction of our microeconomic model. A unified quantitative description in four countries is useful not only from theoretical point of view as a possibility to mathematically describe the process of personal income distribution as the evolution of a physical system. The universal character of personal income evolution as a unique function of real GDP per capita allows accurate forecasting of very specific income characteristics related to fiscal, monetary and other types of socio-economic policies. Extending the observed linear trends of real GDP growth into the future (see Figure 5), one can be use the past U.S. PIDs as templates for the future PIDs in three countries at a time horizon from 15 year (Canada) to 50 years (New Zealand).