Complexity, Statistical physics, Statistics

Victor M. Yakovenko

# Physics-inspired analysis of the two-class income distribution in the USA in
1983–2018

Danial Ludwig and Victor M. Yakovenko  JQI, Department of Physics, University
of Maryland, College Park, Maryland, 20742, USA<EMAIL_ADDRESS>
###### Abstract

The first part of this paper is a brief survey of the approaches to economic
inequality based on ideas from statistical physics and kinetic theory. These
include the Boltzmann kinetic equation, the time-reversal symmetry, the
ergodicity hypothesis, entropy maximization, and the Fokker-Planck equation.
The origins of the exponential Boltzmann-Gibbs distribution and the Pareto
power law are discussed in relation to additive and multiplicative stochastic
processes. The second part of the paper analyzes income distribution data in
the USA for the time period 1983–2018 using a two-class decomposition. We
present overwhelming evidence that the lower class (more than 90% of the
population) is described by the exponential distribution, whereas the upper
class (about 4% of the population in 2018) by the power law. We show that the
significant growth of inequality during this time period is due to the sharp
increase in the upper-class income share, whereas relative inequality within
the lower class remains constant. We speculate that the expansion of the
upper-class population and income shares may be due to increasing digitization
and non-locality of the economy in the last 40 years.

###### keywords:

Econophysics, Economic inequality, Boltzmann-Gibbs distribution, Pareto power
law, Two-class society, Stochastic processes

arXiv:2110.03140v2, 5 January 2022. Submitted to Philosophical Transactions of
the Royal Society A for the special issue “Kinetic exchange models of
societies and economies”.

##  1 Introduction

Inequality is an important, ubiquitous feature of human society. What is the
origin of economic inequality? Is it inevitable or avoidable? Are there common
mathematical patterns of inequality? Is quantitative description of such
patterns possible? These questions have puzzled scientists for centuries.
Lately, in the last 20 years or so, the subject has moved to the center of
public discourse.

Statistical physics and kinetic theory deal with big statistical ensembles
and, as such, are branches of the applied theory of probabilities. The
econophysics movement, emerging about 25 years ago [1, 2], utilizes
mathematical methods developed in these disciplines for probabilistic
description of social ensembles. Several papers in the late 1990s – early
2000s applied these ideas to economic inequality [3, 4, 5, 6]. Since then, the
volume of such literature has grown enormously [7, 8, 9, 10]. Importantly, the
econophysics approach has attracted attention of economists and social
scientists [11, 12] and is now debated within these disciplines [13], as well
as in popular media [14, 15]. Physics-inspired approaches to economic
inequality are briefly reviewed in Sec. 2 of this paper, particularly for the
benefit of non-physicists.

Then, in Sec. 3, these ideas are applied to a quantitative analysis of the
income distribution data in the United States for 1983–2018. We use the
publicly available data from Publication 1304 by the Statistics of Income
research division of the Internal Revenue Service (IRS), the United States tax
agency [16]. The earliest year covered by Publication 1304 is 1983, and the
latest year currently available is 2018, while the earlier data are available
in Publication 79. Our quantitative study of such a wide historical range
reveals salient trends of inequality evolution in the United States. The paper
ends with Conclusions in Sec. 4.

##  2 Physics-inspired approaches to economic inequality

This Section briefly outlines mathematical approaches to economic inequality
introduced in the early econophysics papers [3, 4, 5, 6] and subsequently
followed by numerous further extensions. It is not intended as a comprehensive
literature survey, but rather as a quick summary of the ideas relevant to our
quantitative analysis of income distribution data presented in Sec. 3.

###  2.1 The Boltzmann kinetic equation

One approach is based on the Boltzmann kinetic equation. First, let us
consider an ideal gas, where molecules have kinetic energies $\varepsilon$ and
experiencing pairwise collisions. The total energy of two molecules remains
constant before and after collision

$\varepsilon_{1}+\varepsilon_{2}=\varepsilon_{1}^{\prime}+\varepsilon_{2}^{\prime},$
(1)

while the energy $\varepsilon_{1}-\varepsilon_{1}^{\prime}$ is transferred
from the first to the second molecule.

By analogy, now let us consider a monetary transaction between two economic
agents, whose money balances are $m_{1}$ and $m_{2}$ [4]. Upon transaction,
the first agent transfers some amount of money $\Delta$ to the second agent in
payment for goods or services. Then the money balances of the agents become
$m_{1}^{\prime}=m_{1}-\Delta$ and $m_{2}^{\prime}=m_{2}+\Delta$, so the sum is
conserved

$m_{1}+m_{2}=m_{1}^{\prime}+m_{2}^{\prime}.$ (2)

The conservation of money expressed by Eq. (2) is nothing but the law of
accounting [17]. From the accounting perspective, all monetary transactions
are transfers of digits from one account to another, so the decrease in one
account is equal to the increase in another. The flow of monetary digits,
i.e., the abstract bits of information, between different accounts constitutes
the informational layer of the economy [17]. It is coupled to another,
physical layer of the economy, which represents counterflow of physical goods
and services in transactions between agents. In contrast to the monetary
digital layer, the items in the physical layer are not conserved, because they
can be produced, consumed, and destroyed. However, production of physical
items, by itself, does not change the number of digital monetary units in the
system, because they belong to different layers and are governed by different
rules.

For a statistical ensemble of many interacting economic agents, we introduce
the probability distribution $P(m,t)$ of money $m$ among the agents. Its
evolution as a function of time $t$, as a result of monetary transactions, can
be described by the Boltzmann kinetic equation [18]

$\displaystyle\frac{\partial P(m,t)}{\partial t}=\int\\!\\!\\!\\!\int
dm^{\prime}\,d\Delta\,$
$\displaystyle\left\\{-\Gamma_{[m,m^{\prime}]\to[m-\Delta,m^{\prime}+\Delta]}P(m,t)P(m^{\prime},t)\right.$
(3)
$\displaystyle\left.+\Gamma_{[m-\Delta,m^{\prime}+\Delta]\to[m,m^{\prime}]}P(m-\Delta,t)P(m^{\prime}+\Delta,t)\right\\}.$

Here $\Gamma_{[m,m^{\prime}]\to[m-\Delta,m^{\prime}+\Delta]}$ is the
probability of transferring money $\Delta$ from an agent with money $m$ to an
agent with money $m^{\prime}$ per unit time. This probability, multiplied by
the occupation numbers $P(m,t)$ and $P(m^{\prime},t)$, gives the rate of
transitions from the state $[m,m^{\prime}]$ to the state
$[m-\Delta,m^{\prime}+\Delta]$. The first term in Eq. (3) gives the
depopulation rate of the state $m$, whereas the second term represents the
reversed process, where the occupation number $P(m,t)$ increases.

In order to make further progress, different papers make various assumptions
about the structure of the transition probabilities
$\Gamma_{[m,m^{\prime}]\to[m-\Delta,m^{\prime}+\Delta]}$. For example, Ref.
[5] introduced a model of saving propensity, where the agents save some
fraction of their money balances before engaging in monetary transactions
described by Eq. (2). Whether such specific model assumptions are realistic
remains an open question.

Many papers, particularly in this volume, apply Eq. (3) not to money $m$, but
to assets [3] or wealth $w$. Some papers seem to use $m$ and $w$ synonymously,
in which case there is no difference. But more generally, wealth $w=m+ps$ is a
sum of the money balance $m$ and the monetary value $ps$ of tangible property,
e.g., stocks $s$, where $p$ is the stock price. Then, wealth is generally not
conserved, because it may increase or decrease due to a change in price $p$,
while an agent is not conducting any transactions at all. For example, during
the early COVID crisis in March 2020, many agents stayed passive and performed
no transactions, but trillions of dollars of their wealth first disappeared
and then reappeared due to changes of stock prices. In contrast, money
balances were largely unaffected by such wild swings.

###  2.2 The equilibrium Boltzmann-Gibbs distribution

Interestingly, a stationary solution of Eq. (3) can be found without detailed
knowledge of the transition probabilities
$\Gamma_{[m,m^{\prime}]\to[m-\Delta,m^{\prime}+\Delta]}$, if they satisfy the
time-reversal symmetry

$\Gamma_{[m,m^{\prime}]\to[m-\Delta,m^{\prime}+\Delta]}=\Gamma_{[m-\Delta,m^{\prime}+\Delta]\to[m,m^{\prime}]}.$
(4)

The condition (4) means that the probabilities of transactions in direct and
reversed directions are equal. The fundamental dynamical equations in physics
do satisfy the time-reversal symmetry. In contrast, there is no such general
requirement in economics, e.g., the model [5] with saving propensity does not
have the time-reversal symmetry. Nevertheless, such symmetry may be
approximately applicable in some cases.

In Eq. (3), the probability distribution $P(m)$ is stationary, $dP(m)/dt=0$,
when the rates of direct and reversed transitions are equal, so that the two
terms in the curly brackets cancel each other. This is called the principle of
detailed balance. If the time-reversal condition (4) is satisfied, then the
transition probabilities $\Gamma$ cancel out, and the detailed balance
condition becomes

$P(m)P(m^{\prime})=P(m-\Delta)P(m^{\prime}+\Delta).$ (5)

In the notation of Eq. (2), Eq. (5) has the form
$P(m_{1})P(m_{2})=P(m_{1}^{\prime})P(m_{2}^{\prime})$ and can summarized as
follows. The probability $P$ is multiplicative, whereas the energy
$\varepsilon$ or the money $m$ appearing in the argument of $P$ is additive,
due to conservation laws (1) or (2). A general solution of Eq. (5) is given by
the exponential Boltzmann-Gibbs distribution [4]

$P(m)=c\,e^{-m/T_{m}}.$ (6)

Here the coefficient $c=1/T_{m}$ is obtained from the normalization condition
$\int_{0}^{\infty}dm\,P(m)=1$, whereas the money temperature $T_{m}=\langle
m\rangle=\int_{0}^{\infty}dm\,P(m)\,m$ is equal to the average money balance
of an agent.

We assume that the boundary condition $m\geq 0$ is imposed, so that debt,
which can be interpreted as a negative money balance, is not permitted. Thus,
$m=0$ represents the lowest possible balance of an agent, analogous to the
ground-state energy in physics. Permitting debt, i.e., negative $m$, makes the
problem more complicated and potentially unstable [17], unless a limit on debt
is imposed [4, 19]

The Boltzmann-Gibbs distribution of energy $P(\varepsilon)\propto
e^{-\varepsilon/T_{\varepsilon}}$, analogous to Eq. (6), plays a fundamental
role in the equilibrium statistical physics. In textbooks on this subject
[20], it is usually derived using other methods, without reference to the
dynamical kinetic equation (3). In particular, Eq. (6) can be derived
geometrically from the ergodicity hypothesis [21, 22], which postulates that
all microscopic configurations of the system consistent with macroscopic
constraints are equally probable.

Alternatively, the Boltzmann-Gibbs distribution (6) can be obtained by
maximizing the entropy of the system $S=-\int_{0}^{\infty}dm\,P(m)\,\ln[P(m)]$
with a constraint on the total money in the system $\langle
m\rangle=\int_{0}^{\infty}dm\,P(m)\,m$ using the method of Lagrange
multipliers. The entropy $S=\ln W$ is the logarithm of the multiplicity $W$,
which is the number of microscopic realizations of a given distribution $P(m)$
obtained by combinatorial permutations of the agents. The principle of entropy
maximization follows from the H-theorem proved by Boltzmann [18]. The theorem
shows that, for the probability distribution $P(m,t)$ obeying the Boltzmann
kinetic equation (3), the entropy $S(t)$ monotonously increases in time, until
it reaches its maximal value at the equilibrium Boltzmann-Gibbs distribution
(6). Time evolution of global inequality in energy consumption and $\rm
CO_{2}$ emissions per capita was interpreted in Refs. [23, 24] as a
manifestation of entropy increase due to economic globalization.

###  2.3 The Fokker-Planck equation

When the change $\Delta$ is small for an agent with money balance $m$, the
integro-differential Boltzmann kinetic equation (3) can be reduced the
partial-differential Fokker-Planck equation [18], also known as the diffusion
equation or the Kolmogorov forward equation,

$\frac{\partial P(m,t)}{\partial t}=\frac{\partial}{\partial
m}\left[A(m)P(m,t)+\frac{\partial}{\partial m}B(m)P(m,t)\right],\quad
A=-{\langle\Delta\rangle\over dt},\quad B={\langle\Delta^{2}\rangle\over
2dt}.$ (7)

The coefficients $A$ and $B$, known as the drift and diffusion, are the first
and second moments of the random money balance changes $\Delta$ per time
increment $dt$.

Assuming the boundary condition $m\geq 0$, a stable stationary solution of Eq.
(7) exists only for the drift in negative direction,
$\langle\Delta\rangle/dt<0$, so that the coefficient $A>0$ is positive. The
stationary solution $\partial_{t}P=0$ of Eq. (7) is then obtained by setting
the probability flux, which is the expression in the square brackets, to zero:

$P(m)=\frac{c}{B(m)}\exp\left(-\int_{0}^{m}\frac{A(m^{\prime})}{B(m^{\prime})}\,dm^{\prime}\right),$
(8)

where the coefficient $c$ follows from the normalization condition
$\int_{0}^{\infty}P(m)\,dm=1$.

In the simplest case where the coefficients $A$ and $B$ are constants
independent of $m$, the stationary distribution (8) is exponential
$P(m)\propto e^{-m/T_{m}}$, thus reproducing the Boltzmann-Gibbs formula (6).
Here the money temperature $T_{m}=B/A$ is expressed in terms of $B$ and $A$
similarly to the Einstein relation between temperature, diffusion, and
mobility [18]. A well-known example in physics is the barometric distribution
of gas density $P(z)\propto e^{-\mu gz/T}$ versus the height $z$ in the
presence of gravity, where $\mu$ is the mass of a molecule, $g$ is the
gravitational acceleration, and $\mu gz$ is the gravitational energy of a
molecule. The competition between downward drift due to gravity and spreading
around due to diffusion produces the stationary exponential distribution
$P(z)$. It represents a statistical equilibrium, in contrast to a mechanical
one where two forces acting on an object balance each other (like supply and
demand in economics).

###  2.4 Applications to income distribution

The Boltzmann kinetic equation (3) is a two-body equation, in the sense that
it involves the probability distributions $P(m,t)$ and $P(m^{\prime},t)$ of
two interacting agents, so it is a nonlinear equation. In comparison, the
Fokker-Planck equation (7) is a one-body equation, because it only involves
$P(m,t)$ for one agent, so this equation is linear and sometimes called the
master equation. Coupling between agents is implicitly represented in Eq. (7)
by the first and second moments of the fluctuations $\Delta$ of the variable
$m$ for a given agent, due to its interaction with the “environment”
consisting of the other agents. Unlike the two-body approach, the one-body
description does not invoke a conservation law explicitly. Thus, Eq. (7) can
be generally applied to all kinds of stochastic variables, which is widely
done in the literature. For example, the famous Black-Scholes equation for
random fluctuations of stock prices is based on a version of Eq. (7). A
particular version of Eq. (7) was proposed in Ref. [6] for the distribution of
wealth $w$.

In the rest of the paper, we switch to the probability distribution $P(r)$ of
income $r$, instead of the money balance $m$. The income $r$ of an agent is
the influx of money per unit time, typically per year, so it is analogous to
the power in physics, if money $m$ is analogous to the energy. Following this
definition, the income is non-negative, so it satisfies the boundary condition
$r\geq 0$. The main reason for this change of focus is that empirical data for
income distribution $P(r)$ are available from government agencies, whereas
empirical data for $P(m)$ are difficult to obtain. Income inequality is widely
discussed in the literature and is an important subject by itself.

Let us treat the income $r$ of an agent as a stochastic variable experiencing
random fluctuations $\Delta$ over a time increment $dt$. Then the time-
dependent probability distribution of income $P(r,t)$ obeys the Fokker-Planck
equation (7) with the variable $m$ is replaced by the variable $r$. We focus
on the stationary distribution (8), where, again, $m$ is replaced by $r$. To
make further process, we need to make some assumptions about the drift and
diffusion coefficients $A(r)$ and $B(r)$. Empirical analysis of income
distribution shows that it has a well-defined two-class structure [25, 26].
The lower class is described by the exponential Boltzmann-Gibbs function,
which was uncovered in Ref. [27] and then confirmed for the middle part of
income distributions in many countries [28]. In contrast, the upper class is
described by the Pareto power law [29]. The two-class structure of income
distribution can be rationalized on the basis of a kinetic approach as
follows.

For the lower class, where income comes from wages and salaries, it is
reasonable to assume that income changes do not depend on income itself, i.e.,
$\Delta$ is independent of $r$. Such a stochastic process is called additive
[26]. Then the coefficients in Eq. (7) are some constants $A_{0}$ and $B_{0}$
independent of $r$, and the stationary distribution of income (8) is
exponential, analogous to the Boltzmann-Gibbs distribution (6)

$P_{\rm add}(r)=\frac{1}{T}\,e^{-r/T},\qquad T=\frac{B_{0}}{A_{0}}.$ (9)

Here the parameter $T$ is the income temperature, which could be denoted as
$T_{r}$, but we drop the subscript $r$ to shorten notation. It is equal to the
mean income of the lower class: $T=\langle r\rangle_{\rm add}$.

In contrast, the upper-class income comes from bonuses, investments, and
capital gains, which are calculated as percentages. Therefore, for the upper
class, it is reasonable to expect that income changes are proportional to
income itself, i.e., $\Delta\propto r$. This is known as the proportionality
principle of Gibrat [30], and such a stochastic process is called
multiplicative [26]. Then $A=ar$ and $B=br^{2}$, and Eq. (8) gives a power-law
distribution

$P_{\rm mult}(r)\propto\frac{1}{r^{1+\alpha}},\qquad\alpha=1+\frac{a}{b}.$
(10)

The multiplicative hypothesis for the upper-class income was quantitatively
verified in Ref. [31] for Japan, where tax identification data are officially
published for the top taxpayers.

In practice, the additive and multiplicative processes may coexist. For
example, an employee may receive a cost-of-living raise calculated in
percentages (the multiplicative process) and a merit raise calculated in
dollars (the additive process). Assuming that these processes are
uncorrelated, we find that $A=A_{0}+ar$ and
$B=B_{0}+br^{2}=b(r_{0}^{2}+r^{2})$, where $r_{0}^{2}=B_{0}/b$. Substituting
these expressions into Eq. (8), we find [7, 32]

$P_{\rm
int}(r)=c\,\frac{e^{-(r_{0}/T)\arctan(r/r_{0})}}{[1+(r/r_{0})^{2}]^{(1+\alpha_{i})/2}},$
(11)

The distribution (11) interpolates between the exponential law for low $r$ and
the power law for high $r$, because either the additive or the multiplicative
process dominates in the corresponding limit. A crossover between the two
regimes takes place at $r\approx r_{0}$, where the additive and multiplicative
contributions to $B$ are equal. The distribution (11) has three parameters:
the temperature $T=A_{0}/B_{0}$, the Pareto exponent $\alpha_{i}=1+a/b$, and
the crossover income $r_{0}$. It is a minimal model that captures the salient
features of the two-class income distribution [7, 32]. The label $i$
distinguishes the exponent $\alpha_{i}$ in the interpolated formula (11) from
the exponent $\alpha$ in the simple power law (10) for the purpose of data
fitting, as discussed in Sec. 33.3. A formula similar to (11) was also derived
by Fiaschi and Marsili [33] for a microeconomic model effectively described by
Eq. (7). But Eq. (11) was published for the first time in 1895 in this journal
by the statistician Karl Pearson [34] and is known as the Pearson Type IV
distribution.

Although Eqs. (9), (10), and (11) are derived from the stationary solution (8)
of the Fokker-Planck equation, in the next Section we treat them as quasi-
stationary and empirically determine how their parameters evolve over time.

##  3 Data analysis of income distribution in the USA in 1983–2018

###  3.1 The two-class fit of the cumulative distribution function

Figure 1: The cumulative distribution function $C(r)$ versus the rescaled
income $r/T$ in the USA for 1983–2018. The main panel is in log-log scale,
whereas the inset is in log-linear scale.

We use the variable $r$ for annual income, which is called the adjusted gross
income in the IRS Publication 1304 [16]. Income distribution data are reported
in this publication at fixed income levels. At the high end, these levels are
$100k, $200k, $500k, and $1M for the years earlier than 2000. Here k denotes
thousand dollars, and M million dollars. Starting from 2000, the data for the
income levels of $1.5M, $2M, $5M, and $10M are also reported. At the low end
below $100k, the data are reported at many income levels. Further technical
details about the source and analysis of the data are provided in Supplemental
Material [35].

Given these fixed income bins set by IRS, we introduce the (complementary)
cumulative distribution function
$C(r)=\int_{r}^{\infty}P(r^{\prime})\,dr^{\prime}$ (CDF), obtained by
integration of the probability density function $P(r)$ (PDF). The cumulative
function $C(r)$ is the fraction of tax returns reporting adjusted gross income
above $r$ in the IRS data. In our paper, we treat the fraction of tax returns
as a proxy for the fraction of population with income above $r$. This
correspondence is only approximate for several reasons. First, some tax
returns are filed jointly by married couples, i.e., one tax return per two
people. The fraction of joint tax returns has dropped from about 1/2 in 1983
to about 1/3 in 2018. Second, people with income below a certain threshold are
not required to file tax returns, but many of them do because of refunds from
federal tax withholding and tax credit incentives. Clearly, the correspondence
between the fractions of population and tax returns is not perfect, but it is
the best we can do, given the publicly available IRS data.

The cumulative distribution $C(r)$ is given by the exponential and power-law
functions for the corresponding probability densities (9) and (10):

$C_{\rm add}(r)=e^{-r/T},\qquad\qquad C_{\rm
mult}(r)\propto\frac{1}{r^{\alpha}}.$ (12)

We perform a piecewise fit of the empirical data to $C_{\rm add}(r)$ at the
low end and to $C_{\rm mult}(r)$ at the high end of income distribution. The
IRS publication reports many data points at the low end, which makes it easier
to determine the income temperature $T$ from the fits. In contrast, only few
data points are available at the high end, so the Pareto exponent $\alpha$ is
obtained with lower accuracy.

Having determined the temperature $T$, we visualize the fits in Fig. 1 using
the rescaled income $r/T$ as the dimensionless coordinate on the horizontal
axis. Then the CDF curves for different years collapse on a single curve at
the low end of the distribution. The inset in Fig. 1 shows the data in the
log-linear scale, with the rescaled income $r/T$ in linear scale on the
horizontal axis and CDF in logarithmic scale on the vertical axis. The
collapse of the data points on a straight line in the log-linear scale
indicates that $C(r)$ is, indeed, well described by the exponential function.
For clarity, the data for 1983–1999 and 2000–2018 are separated in the inset.
The data points for the early 1980s deviate slightly below the exponential
line, indicating lower inequality in those years, but this deviation
subsequently goes away. The upper boundary of the exponential fit gradually
shrinks from $r/T\approx 3.5$ in the late 1980s to $r/T\approx 2.5$ by 2000,
with data points at the higher incomes deviating upwards for 2000–2018 in the
inset.

The main panel in Fig. 1 shows the same data in log-log scale, where both axes
for $r/T$ and CDF are logarithmic. At the lower end, the data points collapse
on the black curve representing the exponential distribution. At the high end,
the data points fall on straight lines, representing the power law in log-log
coordinates. The slopes of these lines give the Pareto exponent $\alpha$. We
observe a noticeable decrease in the Pareto slope from 1983 to 1999, which
signals fattening of the upper tail, but then a narrower variation of the
slope in 2000–2018. While the main panel in Fig. 1 presents clear data-based
evidence for the two-class society, it also demonstrates the absence of the
“middle class”. There is no objective, commonly agreed upon definition of the
“middle class” concept in the economic literature, where different authors
make up their own subjective definitions of the “middle class”.

As an alternative to the piecewise fit, in Fig. 2 we show fits of the CDF data
for the last ten years 2009–2018 to the continuous interpolated function (11)
in log-log scale. We first determine $T$ by fitting the low-end data as
described above. Then we determine the Pareto exponent $\alpha_{i}$ and the
crossover income level $r_{0}$ by fitting to Eq. (11). Because of the
different fitting procedure, the value of the Pareto exponent $\alpha_{i}$
obtained from the interpolated fit is slightly different from the value
$\alpha$ obtained from the piecewise fit.

Figure 2: The cumulative distribution function $C(r)$ versus the rescaled
income $r/T$ in the USA for 2009–2018 fitted to the interpolating function
(11). The crossover income $r_{0}$ of the interpolating fit is indicated by
the red star. The small black circle indicates another crossover parameter
$r_{*}$ obtained by piecewise fit.

The crossover point $r_{0}$ between the exponential and power-law
distributions is indicated in Fig. 2 by the red star. An alternative crossover
point $r_{*}$ is obtained as an intersection (indicated by the small black
circle) of the piecewise exponential and power-law fits shown by black curves
for 2018. The values of $r_{0}$ and $r_{*}$ are generally different, but
rather close. We perform the piecewise fit and determine $r_{*}$ for all
available years. In contrast, the interpolated fit is performed only for the
last ten years 2009–2018, where it works well. For the earlier years, it does
not work well, because the crossover between the exponential and power-law
distributions is less smooth and more cuspy, as seen in Fig. 1. Moreover, the
lower part deviates downward from the exponential fit in the early 1980s.

Table 1: Parameters of the fit for 2018. Notation is explained in the text. $r_{\rm med}$ | $T$ | $\langle r\rangle$ | $r_{0}$ | $r_{*}$ | $\alpha$ | $\alpha_{i}$ | $f_{p}$ | $f_{r}$ | $f_{t}$ | $f_{L}$ | G | Gtax
---|---|---|---|---|---|---|---|---|---|---|---|---
$41k | $57k | $78k | $167k | $185k | 1.42 | 1.25 | 4% | 34% | 58% | 22% | 60% | 82%

The two-class decomposition allows us to characterization the empirical income
distribution by a small set of fitting parameters. Their values are shown in
Table 1 for the latest year 2018 and discussed in subsequent sections, where
we visualize the historical evolution of these parameters.

###  3.2 Divergence of the mean and the median incomes

Figure 3: The median $r_{\rm med}$ and mean $\langle r\rangle$ incomes and the
income temperature $T$, which is the mean income of the lower class, in
nominal dollars (top panel) and divided by inflation (bottom panel).

The dotted curves in the top panel of Fig. 3 shows the historical evolution of
the income temperature $T$, as well as the mean $\langle
r\rangle=\int_{0}^{\infty}dr\,r\,P(r)$ and median $r_{\rm med}$ incomes. All
of them increase in time in nominal dollars, but some of that increase is due
to inflation. We define the inflation parameter as the ratio
CPI(year)/CPI(1983) of the Consumer Price Index (CPI) for a given year to the
CPI in 1983 [36]. The solid curves in the top panel of Fig. 3 show the
parameters $T$ and $r_{\rm med}$ taken in 1983 and multiplied by the inflation
parameter for the subsequent years. The solid curves are close to the dotted
curves for $T$ and $r_{\rm med}$, so the increase in nominal dollars of the
income temperature $T$ and the median income $r_{\rm med}$ is primarily due to
inflation.

The bottom panel of Fig. 3 shows the parameters $\langle r\rangle$, $T$, and
$r_{\rm med}$ divided by the inflation parameter. Again, we observe that the
parameters $T$ and $r_{\rm med}$ characterizing the lower class remain
approximately constant over 36 years, when adjusted for inflation. For the
exponential distribution in Eq. (12), these two parameters are simply
proportional: $r_{\rm med}=T\ln 2\approx 0.7\,T$.

In contrast, the overall mean income $\langle r\rangle$ increases sharply
relative to inflation. The divergence between the income temperature
$T=\langle r\rangle_{\rm add}$, which is the mean income of the lower class,
and overall mean income $\langle r\rangle$, which includes the upper class,
points to a significant increase of inequality in the last 36 years. As the
bottom panel of Fig. 3 shows, these two mean incomes were approximately equal
in 1983, but have diverged since then. Thus, the growth of inequality is
primarily driven the increase of the upper-class income relative to the lower
class, while the lower-class income remains approximately constant when
adjusted for inflation.

Figure 4: Top two curves: the Pareto exponents $\alpha$ from the piecewise
fits and $\alpha_{i}$ from the interpolated fits. Bottom two curves: the
crossover incomes $r_{*}$ (from the piecewise fits) and $r_{0}$ (from the
interpolated fits) separating the lower and upper classes, normalized to the
income temperature $T$.

###  3.3 The Pareto exponent and the boundary between two classes

The top two curves in Fig. 4 show the historical evolution of the Pareto
exponents $\alpha$, obtained by the piecewise fits for all years, and
$\alpha_{i}$, obtained by the interpolated fits for the last 10 years using
Eq. (11). The values of $\alpha$ and $\alpha_{i}$ are slightly different
because they are obtained by different fitting procedures, but they exhibit
similar behavior. We observe a significant decrease of the exponent $\alpha$
from 1983 to 2000, which indicates fattening of the tail. But after 2000, the
exponent $\alpha$ exhibits a “bumpy plateau”, where it decreases during
financial bubbles and increases during crashes, but stays approximately
constant overall. These observations are qualitatively consistent with the
main panel in Fig. 1, as well as Fig. 3, and indicate that the sharp expansion
of the upper class happened in the 1980s and 1990s, but have slowed down or
saturated after 2000.

The two curves at the bottom of Fig. 4 show the historical evolution of the
crossover incomes $r_{*}$ and $r_{0}$ separating the lower and upper classes,
as illustrated in Fig. 2. The former is obtained from the piecewise fits for
all years, while the latter from the interpolated fits for the last 10 years,
and both are normalized to the income temperature $T$. The ratio $r_{*}/T$ has
decreased from about 5 in 1983 to slightly above 3 in 2018. A particularly
sharp drop is visible around 1986, coinciding with the Tax Reform Act of the
Reagan administration. The decreasing value of the ratio $r_{*}/T$ indicates
that the exponential part of the distribution is shrinking, whereas the power-
law part is expanding, which is consistent with the inset in Fig. 1.

###  3.4 The income, population, and tax fractions of the upper class

Given the boundary $r_{*}$ between the two classes, we can also characterize
the upper class by the fractions of its population $f_{p}$, income $f_{r}$,
and tax paid $f_{t}$ relative to the total population:

$f_{p}=\int_{r_{*}}^{\infty}dr\,P(r),\qquad f_{r}=\frac{1}{\langle
r\rangle}\,\int_{r_{*}}^{\infty}dr\,r\,P(r),$ (13)

whereas $f_{t}$ is defined similarly to $f_{r}$ but for the tax paid rather
than income. The historical evolution of these fractions is shown in the top
panel of Fig. 5. All of these fractions increase in time, indicating expansion
of the upper class. In particular, the lower curve in the top panel shows that
the upper-class population share $f_{p}$ has increased from less than 1% in
1983 (when almost the whole income distribution was close to exponential) to
slightly greater than 4% in 2018. The increase in the upper-class population
fraction is closely related to the decrease in the crossover income ratio
$r_{*}/T$ separating the two classes, as shown by the bottom curves in Fig. 4.

Figure 5: Top panel: the fractions of upper-class population $f_{p}$, income
$f_{r}$, and tax paid $f_{t}$ relative to the total population. The income
fraction $f_{r}$ is deduced directly from the IRS data, whereas
$f_{r}^{\prime}$ is calculated from Eq. (14). Bottom panel: the ratios of
these fractions.

The income fraction $f_{r}$ of the upper class exhibits the “bumpy plateau”
pattern in the top panel of Fig. 5 for the last 20 years, similarly to the
exponent $\alpha$ in Fig. 4 and the mean income $\langle r\rangle$ in Fig. 3.
We observe sharp peaks of the upper-class income share $f_{r}$ around 2000 and
2017 coinciding with the stock market and real estate bubbles, followed by
crashes of these bubbles, and then further increase by 2012, under the
quantitive easing policy of the Federal Reserve Bank. Clearly, the upper-class
income share is closely related to financial bubbles.

The income fraction $f_{r}$ shown in Fig. 5 is deduced directly from the IRS
data, given the income threshold $r_{*}$. As a consistency check, we compare
it with the theoretically expected fraction $f_{r}^{\prime}$ for the power-law
distribution (10). The expected mean income of the upper class and the income
fraction relative to the total population are

$\langle r\rangle_{\rm
mult}=\frac{\int_{r_{*}}^{\infty}dr\,r/r^{1+\alpha}}{\int_{r_{*}}^{\infty}dr/r^{1+\alpha}}=\frac{\alpha}{\alpha-1}\,r_{*},\qquad\qquad
f_{r}^{\prime}=f_{p}\,\frac{\langle r\rangle_{\rm mult}}{\langle
r\rangle}=f_{p}\,\frac{\alpha}{\alpha-1}\,\frac{r_{*}}{\langle r\rangle}.$
(14)

The upper-class income nominally diverges, $f_{r}^{\prime}\to\infty$, when
$\alpha\to 1$. When extrapolated from the decreasing trend in 1980s and 1990s
in Fig. 4, the exponent $\alpha$ could have reached 1 by 2018, but saturated
after 2000 at the “bumpy plateau” of bubbles and crashed, thus avoiding the
singularity.

The expected income fraction $f_{r}^{\prime}$ calculated from Eq. (14) with
the parameters from the piecewise fits is shown in the top panel of Fig. 5
together with $f_{r}$. We see that $f_{r}$ and $f_{r}^{\prime}$ are very close
for most years, indicating that the power law (10) accounts for all of the
upper-class income. For the recent years, we notice that
$f_{r}>f_{r}^{\prime}$, suggesting that the actual upper-class income may be
slightly higher relative to the power-law extrapolation. The difference
between $f_{r}$ and $f_{r}^{\prime}$ could indicate the additional income of
oligarchy on top of the power law, as proposed for the wealth distribution in
Ref. [37]. However, the difference in Fig. 5 is too small to be taken as
evidence for oligarchy in income distribution, given the uncertainty of the
fitting parameters.

The ratios of the fractions are shown in the bottom panel of Fig. 5. The
middle curve shows the ratio $f_{r}/f_{p}$ of the income and population
fractions. This ratio stays approximately constant $\approx 10$ and even
slightly decreases over time. Comparison with Eq. (14) for
$f_{r}^{\prime}/f_{p}$ suggests that the decrease in $r_{*}/\langle r\rangle$
is compensated by the increase in $\alpha/(\alpha-1)$, as shown in Fig. 4.

Another indicator is the share of taxes $f_{t}$ paid by the upper class,
relative to its income share $f_{r}$. The lower curve in the bottom panel of
Fig. 5 shows a noticeable decrease in $f_{t}/f_{r}$ after the tax reform of
1986, but then it stays approximately constant $\approx 2$. Relatedly, the top
panel shows that, after 2000, the income share $f_{r}$ oscillates roughly
between 20% and 30%, whereas the tax share $f_{t}$ between 40% and 60%.
Notably, the upper class paid close to 60% of taxes in 2018. The top curve in
the bottom panel shows the ratio $f_{t}/f_{p}$ of tax to population fractions
for the upper class. It exhibits a sharp drop around 1986 and then a gradual
decrease in the last 10 years.

###  3.5 Lorenz curves and Gini coefficient

Figure 6: The IRS data points for the Lorenz curves of income distribution in
1983 (red squares) and 2018 (blue circles). The solid and dashed curves
represent the exponential, Eq. (16), and rescaled exponential, Eq. (17),
distributions characterizing the lower class. The population $f_{p}$ and
income $f_{r}$ fractions mark the boundary of the upper class in 2018. The
inset shows the empirical Gini coefficient in comparison with Eq. (18).

Income distribution is commonly characterized by the Lorenz curve in the
economic literature [38]. The Lorenz curve is defined parametrically in terms
of the two coordinates $x(r)$ and $y(r)$

$x(r)=\int_{0}^{r}dr^{\prime}P(r^{\prime}),\quad y(r)=\frac{1}{\langle
r\rangle}\,\int_{0}^{r}dr^{\prime}r^{\prime}P(r^{\prime}).$ (15)

Here $x(r)$ is the fraction of the population with incomes below $r$, and
$y(r)$ is the total income of this population, as a fraction of the total
income in the system. When the parameter $r$ changes from 0 to $\infty$, the
variables $x(r)$ and $y(r)$ change from 0 to 1, thus producing the Lorenz
graph in the $(x,y)$ plane. Its advantage is that all available data are
represented within a finite area in the $(x,y)$ plane, whereas the upper tail
at $r\to\infty$ is inevitably truncated in Figs. 1 and 2.

It was derived in Ref. [27] that the Lorenz curve for the exponential
distribution (9) is

$y=x+(1-x)\ln(1-x).$ (16)

When the upper class is present, the Lorenz curve for the lower class is
obtained [26] by multiplying Eq. (16) by the factor $(1-f_{L})<1$:

$y=(1-f_{L})[x+(1-x)\ln(1-x)].$ (17)

The factor $(1-f_{L})$ is the fraction of total income contained in the
extrapolated exponential distribution, whereas $f_{L}$ is the additional
fraction of income in the upper class. (The subscript $L$ alludes to the
Lorenz curve.)

The red squares and blue circles in Fig. 6 show the IRS data points for the
Lorenz curves of income distribution in 1983 and 2018. The straight diagonal
line represents the reference case of perfect equality, so inequality is
higher when the data points are further away from it. We observe that the
Lorenz curve for 1983 is very close to the solid black curve representing Eq.
(16) for the exponential distribution. Moreover, the data points in the middle
are slightly above the solid black curve, indicating lower inequality than in
the exponential distribution. These conclusions are consistent with the CDF
graphs in Fig. 1 and the observation that $\langle r\rangle\approx T$ for 1983
in Fig. 3.

In contrast, the Lorenz curve for 2018 is shifted down, signaling an increase
in inequality. Most of its data points, except at the high end, are well
fitted by the rescaled exponential formula (17) shown by the dashed black
curve. This indicates that income distribution within the lower class remains
exponential, whereas the increase in inequality comes from the excess share
$f_{L}$ of the upper-class income, determined by the intersection of the fit
(17) with the vertical axis.

Figure 7: The Lorenz curves of tax distribution in 1983 (red squares) and 2018
(blue circles). The black solid curve represents the exponential
distributions, Eq. (16), for a reference. The population $f_{p}$ and tax
$f_{t}$ fractions of the upper class are indicated for 2018. The inset shows
the tax Gini coefficient.

Also shown in Fig. 6 are the population $f_{p}$ and income $f_{r}$ fractions
of the upper class. Both parameters $f_{L}$ and $f_{r}$ represent an income
share of the upper class, but they are defined differently. In Fig. 6, the
parameter $f_{r}=34\%$ is the income share of the top 4% of population, whose
income distribution follows a power law. In contrast, $f_{L}=22\%$ is the
extra income of the upper class relative the exponential distribution
extrapolated to $r\to\infty$. Thus, the parameter $f_{L}$ measures a deviation
of the actual income distribution from the purely exponential one.

To characterize inequality by a single number, the economic literature [38]
uses the Gini coefficient $0\leq G\leq 1$, defined as twice the area between
the diagonal line and the Lorenz curve. It was shown that $G=1/2$ for the
exponential distribution [27], and

$G=\frac{1+f_{L}}{2}$ (18)

when taking into account the fraction $f_{L}$ of the upper-class income on top
of the exponential distribution [26]. The values of $G$ deduced from the IRS
data are shown by the open circles in the inset of Fig. 6, where the crosses
show the values calculated by Eq. (18) starting from 1988. The agreement
between the empirical values of $G$ and the formula (18) in Fig. 6
demonstrates that the increase in income inequality starting from the late
1980s comes completely from the growth of the upper income share relative to
the lower class, whereas income inequality within the lower class itself
remains constant and exponentially distributed.

Besides the overall increase since 1983, the inset in Fig. 6 exhibits local
maxima corresponding to bubbles in financial markets. The Gini coefficient $G$
peaks around 1988 during the bubble in Savings and Loan, around 2000 in the
stock market, around 2007 in subprime mortgages, and past 2012 under
quantitative easing. Thus we confirm that the overall income inequality, as
measured by the Gini coefficient, peaks during bubbles in financial markets.
This behavior corresponds to the “bumpy plateau” already discussed for Figs.
3, 4, and 5.

Figure 8: Income and tax shares of the bottom 50% and the top 1% of
population.

The IRS publication [16] also provides information on the distribution of
taxes paid by the population. In Fig. 7 we show the Lorenz curves for tax
revenue in 1983 and 2018, along with the Gini coefficient for taxes in the
inset. Because of progressive taxation, where the recipients of higher income
are subject to higher tax rates, the inequality of tax distribution in Fig. 7
is higher than in Fig. 6 for income distribution, as also exemplified by the
higher tax Gini coefficient. We observe that the tax revenue became highly
concentrated in the upper class by 2018, because of the relative decrease in
the lower-class income due to the overall increase in income inequality. As
shown in Fig. 7, the upper class, consisting of only about $f_{p}=4\%$ of
population, contributed the fraction $f_{t}=58\%$, greater than a half, of the
total tax revenue, as also shown by the top curve in Fig. 5.

To amplify these observations, we show the income and tax shares of the bottom
50% and the top 1% of population in Fig. 8, as derived from the Lorenz curves
for income and taxes. From 1983 to 2018, the income share of the lower half of
population has decreased from about 18% to 12%, whereas the income share of
the top 1% of population has doubled from about 9% to 21%. At the same time,
the sum of their income shares remained approximately constant, increasing
only slightly from 27% in 1983 to 33% in 2018. For this reason, the two middle
curves in Fig. 8 look somewhat like mirror images of each other. Thus, the
income share increase of the top 1% of population mostly came from the income
share decrease of the bottom 50% of population.

During the same years, the tax share of the bottom 50% of population has
decreased from about 7% to 3%. In 2018, the total tax contribution of the
lower half of the population is almost negligible, because their total income
share is so low. In contrast, , the tax share of the top 1% of population has
doubled from about 20% to 40%. Thus, the majority of tax revenue now comes
from the top few percent of population, where most of income is concentrated.

A proposal for increasing taxes on the top 1% of population and incomes above
approximately $500k is currently debated by the Democratic Party of the United
States [39]. The quantitative characterization of the two-class structure of
income distribution presented in our paper could provide useful input for the
formulation of a tax policy. In particular, it would be reasonable to align
tax brackets with the boundary between the two classes.

##  4 Conclusions

In this paper, income distribution data for the USA in 1983–2018 is analyzed
in terms of a two-class decomposition, where the lower class is described by
an exponential function, and the upper class by a power law, as inspired by
statistical physics. The quantitative characterization of the two classes
presented here may be useful for the formulation of a tax policy. We observe a
sharp increase of income inequality from the early 1980s to the late 1990s,
but slower increase or saturation after 2000. The inequality pattern after
2000 exhibits a “bumpy plateau” pattern, with ups and downs related to
financial bubbles and crashes, but an approximately constant overall level.

The two-class analysis shows that the growth of inequality originates
primarily from the increasing share of income ($f_{r}$ or $f_{L}$) in the
upper class, relative to the lower class. At the same time, relative
inequality within the lower class, encompassing more than 90% of population,
remains constant and exponentially distributed. Moreover, the mean income $T$
of the lower class and the median income $r_{\rm med}$ remain approximately
constant over 36 years when adjusted for inflation, whereas the overall mean
income $\langle r\rangle$ increases relative to inflation. The growing
divergence between the mean and median incomes is another manifestation of
increasing inequality.

Not only does the upper-class income share $f_{r}$ increase, but the fraction
$f_{p}$ of population belonging to the upper class increases too. In a
relative sense, the upper-class population expands (while still remaining a
small fraction of only about 4% of the total population), while the lower-
class population shrinks (while still remaining the overwhelming majority).
This happens because the ratio $r_{*}/T$ of the crossover income $r_{*}$
separating the two classes to the mean income of the lower class $T$ has
significantly decreased from 1983 to 2018. Thus, the relative income level
threshold for getting into the upper class has decreased.

These empirical observations prompt a descriptive interpretation. We speculate
that the significant increase in the upper-class population fraction $f_{p}$,
as well as their income fraction $f_{r}$, is due to digitization of the
economy in the last 40 years. There was a rapid proliferation of personal
computers during the 1980s, followed by the spread of the Internet and the
World Wide Web in the 1990s, and then by ubiquitous personal mobile devices
and a shift to cloud computing at the present time. This transformation
enabled the creation and relatively easy scaling-up of digital platforms for
highly non-local business operations. In the past, many business were local.
For example, each town had a few local taxi companies, several local book
stores, video rental stores, etc. Now they are largely displaced by a small
number of national and global network platforms, such as Uber and Lyft for
riding, Amazon for books initially and then for all kinds of goods, Netflix
for DVD rentals and video streaming, etc. The founders and owners of such
network platforms become super-rich, because these platforms serve a huge
number of customers, in contrast to the old-fashioned local businesses. Thus
we speculate that the growth of the upper class is due to the ongoing
transformation of business network topology from local clusters to highly
connected superclusters and global hubs. In contrast to the one-to-one
interaction exemplified by Eq. (2), networked businesses are characterized by
the one-to-many interaction topology, resulting in the power law distribution
for the upper class.

Supporting data are accessible in [35].

DL performed data analysis and produced the data-based figures under the
guidance of VY, then VY wrote the manuscript. All authors proofread and
approved the manuscript.

The authors declare that they have no competing interests.

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