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*^{a}*K***, where Y is the measure of production (**^{b}*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*) −

**α**

*M*(

*t*) /

*Λ*(t) (4)

where

*M*(*t*) is the rate of money income denominated in dollars per year [$/*y*],*t*is the work experience expressed in years [*y*];*σ*(*t*) is the capability to earn money, which is a permanent feature of an individual [$/*y*^{2}];*Λ*(*t*) is the size of the earning means, which is a permanent income source of an individual [$/*y*]; and*α**is the dissipation factor [$/**y*^{2}].
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*τ*_{0},*i.e.*her work experience is*t*_{0}= 0. By year*τ*this person will have*t*=*τ*−*τ*_{0}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*(*τ*_{0},*t*) and*Σ*(*τ*_{0},*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*(

*τ*

_{0}

*,t*) =

*Σ*(

*τ*

_{0}

*,t*

_{0}) [

*Y*(

*τ*) /

*Y*(

*τ*

_{0}) ]

^{1/2}(6)

*Λ*(

*τ*

_{0}

*,t*) =

*Λ*(

*τ*

_{0}

*t*

_{0}) [

*Y*(

*τ*) /

*Y*(

*τ*

_{0}) ]

^{1/2}(7)

where

*Σ*(*τ*_{0}*,t*_{0}) and*Λ*(*τ*_{0}*,t*_{0}) are the initial values of capability and instrument for a person with zero work experience in year*τ*_{0};*Y*(*τ*_{0}) and*Y*(*τ*) are the aggregate output per capita values in the years*τ*_{0}and*τ*, respectively, and*dY*(*τ*_{0}*,t*)=*Y*(*τ*)*/Y*(*τ*_{0})=*Y*(*τ*_{0}+*t*)*/Y*(*τ*_{0}) is the cumulative output growth. Note that the initial values*Σ*(*τ*_{0}*,t*_{0}) and*Λ*(*τ*_{0}*,t*_{0}) depend only on the year when the person turns 15,*τ*_{0}*,*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*(*τ*_{0}) and*Σ*(*τ*_{0}), respectively. The product of equations (6) and (7),*Σ*(*τ*_{0}*,t*_{0})*Λ*(*τ*_{0}*,t*_{0}), 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*(*τ*_{0}*,t*_{0})*Σ*(*τ*_{0}*,t*_{0}) is the initial capacity.
Equation
(4) can be re-written to account for the dependence on the initial year,

*τ*_{0}:*dM*(

*τ*

_{0}

*,t*) /

*dt*=

**α**{

*Σ*(

*τ*

_{0}

*,t*) −

*M*(

*τ*

_{0}

*,t*) /

*Λ*(

*τ*

_{0}

*,t*)} (8)

Note
that when we fix

*τ*_{0}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*τ*_{0}only through the influence of the latter on the initial earning capability and instrument. In other words,*τ*_{0}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*(*τ*_{0}*,t*), and the size of earning means,*Λ*(*τ*_{0}*,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*τ*_{0}: min*Σ*(_{k}*τ*_{0}*,t*)=Σ*(*_{min}*τ*_{0}*,t*) and min*Λ*(_{k}*τ*_{0}*,t*)=*Λ*(_{min}*τ*_{0}*,t*), respectively, where*Σ*(_{k}*τ*_{0}*,t*) and*Λ*(_{k}*τ*_{0}*,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}*τ*

_{0}

*,t*) =

*Σ*(

_{k}*τ*

_{0}

*,t*) / Σ

*(*

_{min}*τ*

_{0}

*,t*) (9)

and

*L*(

_{k}*τ*

_{0}

*,t*) =

*Λ*(

_{k}*τ*

_{0}

*, t*) /

*Λ*(

_{min}*τ*

_{0}

*,t*) (10)

where

*S*(_{k}*τ*_{0}*,t*) and*L*(_{k}*τ*_{0}*,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*and_{k}*L*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_{k}*S*(_{k}*τ*_{0}*,t*_{0}) and*L*(_{k}*τ*_{0}*,t*_{0}), for any*τ*_{0}and*t*_{0}, 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}*τ*_{0}*,t*_{0}) and*L*(_{j}*τ*_{0}*,t*_{0}):*S*_{1}(*τ*_{0}*,t*_{0})=2*, . . . , S*_{29}(*τ*_{0}*,t*_{0})=30, and similarly for*L*(_{j}*τ*_{0}*,t*_{0}), 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}*S*_{29}=*L*=_{max}*L*_{29}=30 is only 15 times larger than the smallest*S*=_{min}*S*_{1}=*L*=_{min}*L*_{1}=2. In the model, the minimum values*Σ*and_{min}*Λ*are found to be two times smaller than the smallest possible values of_{min}*L*_{1}and*S*_{1}, respectively. Because the absolute values of variables*Σ*,_{i}*Λ*,_{j}*Σ*and_{min}*Λ*evolve with time according to the same law described in (6) and (7), the relative and dimensionless variables_{min}*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*is constant over all 29 discrete values of the size and the same is valid for_{j}*S*. In a given year_{i}*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*and_{i}*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*combinations define a unique time history._{i}L_{j}
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*and_{i}*L*. The PDF is well approximated by an exponential function 0.033exp(-2.9_{j}*pc*) 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(-2.9*pc*) 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*and_{i}*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*(*τ*_{0}*,t*) and*Λ*(*τ*_{0}*,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*τ*_{0}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( -

**α**

*t*/

*Λ*) ] (11)

Equation
(11) indicates that personal income growth in the absence of economic growth,
i.e. d

*Λ*/d*t*=d*Σ*/d*t*=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*and_{i}*L*and the time dependent minimum values_{j}*Σ*and_{min}*Λ*for_{min}*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*and_{max}*Λ*in a given calendar year,_{max}*τ,*for a given work experience,*t*. The normalized equation for the rate of income,*M*(_{ij}*t*), of a person with capability,*S*, and the size of earning means,_{i}*L*, where_{j}*i, j*={2*, . . . ,*30} is as follows:*M*(

_{ij}*t*) / [

*S*] =

_{max}L_{max}*Σ*(

_{min}Λ_{min}*S*)(

_{i}/S_{max}*L*){1 – exp( –

_{j}/L_{max}

**α**

*t/*[(

*Λ*)(

_{min}L_{max}*L*)]}

_{j}/L_{max}*(12)*

or
compactly:

*M̃̃*(

_{ij}*t*) =

*Σ*[1 – exp ( −

_{min}Λ_{min}S̃_{i}L̃_{j}*t*(1/

*Λ*)(α̃/

_{min}*L̃*))] (13)

_{j}
where

*M̃*(

_{ij}*t*) =

*M*(

_{ij}*t*) / (

*S*)

_{max}L_{max}*S̃*=

_{i}*S*

_{i }/ S_{max}*L̃*=

_{j }*L*

_{j }/ L_{max}**α̃**

*=*

**α**

*/ L*

_{max}
and

*S*=_{max}*L*=30. In this representation,_{max}*S̃*and_{i}*L̃*range from 2/30 to 1. The modified dimensionless dissipation factor_{j}**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}*τ*_{0}*,t*)*Λ*(_{min}*τ*_{0}*,t*) then corresponds to the total (cumulative) growth of real GDP per capita from the start point of a personal work experience,*τ*_{0}(*t*_{0}=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}*τ*_{0}*,t*)*Λ*(_{min}*τ*_{0}*,t*), for each particular starting year*τ*_{0}. Then equation (13) becomes dimensionless and the coefficient changes from*Σ*(_{min}*τ*_{0}*,t*_{0})*Λ*(_{min}*τ*_{0}*,t*_{0})=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*and_{j}*S*, one can derive from (13) the time needed to reach the absolute income level H, where H<1: i="">_{i}*t*

_{H}

*= Λ*ln[1-H)] /

_{j}**α**(14)

This equation is correct only for persons capable
to reach H, i.e. when

*L*H. With all other terms in (14) being constant, the size of work instrument available for a person,_{j}S_{i}/S_{max}L_{max}>*Λ*, defines the change in_{j}*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*. We consider two different values of_{j}*S*=2 and 30 and same value of_{i}*L*=30 and compare personal incomes curves in 1930 and 2011. For constant_{j}*Λ*, the time needed to reach H_{j}*S*for a given person does not depend_{j}L_{i}*S*– the curves in the left and right panels are pair-wise identical in terms of shape. This means that the person with_{i}*S*=2 reaches, say, 50% of her maximum personal capacity_{i}*Λ*_{j}*Σ*exactly at the same time as the person with_{i }*S*=30 reaches 50% of her maximum income. At the same time, the person with_{i}*S*=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._{i}
The increase in

*Λ*from 1930 to 2011 results in a much slower income growth. Solid lines in Figure 4 represent the solutions for constant_{j }*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*=30. The increase in_{j}*Λ*from 1930 to 2011 results in slower income growth. Solid lines represent the solutions for constant_{j }*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*

_{i}Λ_{j}**α**

*t*/

*Λ*

_{j}= Σ_{i}**α**

*t*(15)

*i.e*. the money income,

*M*, for a given person is a linear function of time since

*Σ*and

_{i }**α**are both constants. Using (5), one regains the original meaning of the personal capability to earn money:

*M*(

_{ij}*t*)=

*σ*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

_{i}t.**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.

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