# The fate of the American dream: A first passage under resetting approach to
income dynamics

Petar Jolakoski1 1Research Center for Computer Science and Information
Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2,
1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University Arnab Pal2111
<EMAIL_ADDRESS>1Research Center for Computer Science and Information
Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2,
1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University Trifce Sandev1,3,4
1Research Center for Computer Science and Information Technologies, Macedonian
Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University Ljupco Kocarev1,5
1Research Center for Computer Science and Information Technologies, Macedonian
Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University
Ralf Metzler3,6 1Research Center for Computer Science and Information
Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2,
1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University Viktor
Stojkoski7,8222Corresponding author<EMAIL_ADDRESS>1Research Center for Computer Science and Information Technologies, Macedonian
Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India
3Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-
Golm, Germany
4Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril
and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
5Faculty of Computer Science and Engineering, Ss. Cyril and Methodius
University, PO Box 393, 1000 Skopje, Macedonia
6 Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
7Center for Collective Learning, ANITI, University of Toulouse
8Faculty of Economics, Ss. Cyril and Methodius University

###### Abstract

Detailed knowledge of individual income dynamics is crucial for investigating
the existence of the American dream: Are we able to improve our income status
during our working life? This key question simply boils down to observing
individual status and how it moves between two thresholds: the current income
and the desired income. Yet, our knowledge of these temporal properties of
income remains limited since we rely on estimates coming from transition
matrices which simplify income dynamics by aggregating the individual changes
into quantiles and thus overlooking significant microscopic variations. Here,
we bridge this gap by employing First Passage Time concepts in a baseline
stochastic process with resetting used for modelling income dynamics and
developing a framework that is able to crucially disaggregate the temporal
properties of income to the level of an individual worker. We find
analytically and illustrate numerically that our framework is orthogonal to
the transition matrix approach and leads to improved and more granular
estimates. Moreover, to facilitate empirical applications of the framework, we
introduce a publicly available statistical methodology, and showcase the
application using the USA income dynamics data. These results help to improve
our understanding on the temporal properties of income in real economies and
provide a set of tools for designing policy interventions.

## Introduction

How long does it take for individuals to improve their income profiles within
the socio-economic ladder? For James Truslow Adams, this question formed the
bedrock for the existence of the American dream [1]. If the time is comparable
to the working life of an individual, then it can be argued that social and
economic systems function as envisaged. Otherwise, there might be a need to
reorganize socio-economic policies. As a result, developing methods for
studying the temporal properties of income dynamics is a fundamental question
in the literature of economic inequality and mobility [2, 3, 4, 5, 6, 7, 8, 9]
(Fig. 1(a-b)).

These temporal changes are usually studied by using income transition matrices
[9, 8, 10, 11, 12]. An income transition matrix aggregates and summarizes the
probability of a worker to move between two arbitrary income quantiles, $k$
and $l$, within a given time period (Fig. 1(c)). Using this matrix, we can
estimate the expected time for a worker to first reach the lowest income that
is required to reach $l$ given that the worker is currently in quantile $k$.
This quantity is known as the mean first passage time (MFPT) [13, 14, 15] and
is an adequate approximation for the time required for a worker to improve
their income level [14, 13].

Transition matrices, however, only provide aggregated quantities for the time
properties of income. That is, because of the aggregation we are unable to
differentiate the fortunes of workers that are members of the same quantile.
Indeed, empirical observations of income dynamics exhibit a fractal like
structure [16], implying that not everyone within the quantile will have the
same probability to change their income status. Instead, there are intra-
quantile dependencies that govern the MFPT. As a result, in order to move away
from this drawback of transition matrices, the more recent literature has
developed alternate measures for tracking income fluctuations that are able to
disaggregate the transition matrices to a higher resolution [17]. But, despite
these innovations, our knowledge about the time at which these fluctuations
occur remains limited.

To bridge this gap, we build an analytical framework for disaggregating the
MFPT to the level of an individual worker in the economy. We exploit the
properties of an established stochastic process used for modelling income
dynamics called Geometric Brownian Motion with stochastic resetting (srGBM)
[18, 19]. srGBM has been widely used for investigating the role of various
phenomena on income dynamics: from taxes [20] up to changes in skill prices
[21]. Mathematically, our srGBM-MFPT framework can be seen as a natural
generalization of the finite-state transition matrix approach to processes
with a continuum of states (Fig. 1(d)) [21]. Practically, it can be seen as an
answer to whether the American Dream is a reality for a particular worker. We
utilize our analytical results to develop a statistical methodology for
applying the MFPT in real-world data and display its application by providing
estimates for the time properties of the income distribution in the United
States (USA) for the period between 1978-2015. With these estimates we are
able to see a more granular and comprehensive picture than previous approaches
for the ability of a worker to move across the income distribution. Hence,
they can be used by policy makers to answer important questions, such as: What
is the time needed for a currently minimum wage worker to spend in the
workforce in order to reach a reasonable income level? Which proportion of
workers are able to reach the highest status during their working life? How
easy it is for workers to reach certain income targets? How can we recommend
policies for optimizing the time required for a worker to improve their status
in an economy? To support research and policy interventions in the pursuit of
the answers to these questions, we release a simplified notebook for
calculating the srGBM MFPT by using real world data at:
https://colab.research.google.com/drive/1quV4bdaNaGVmUB1EA0XOINqWC93Y8JBX.

## Results

To develop a “disaggregated” view for the MFPT across each worker of the
income distribution, we use the properties of geometric Brownian motion with
stochastic resetting (srGBM) [18, 22, 19, 23, 21, 24, 25, 26] which is
pertinent to income dynamics. For example, this model confers several real
world socio-economic phenomena such as the power law stationary income
distributions [21, 27, 20] or the famous Great Gatsby curve that visualizes
the relationship between inequality and mobility [19]. In our approach, we
simply assume that time is continuous and there is a population of $N$ workers
in the economy. The income $x(t)$ of a worker in period $t$ grows
multiplicatively with a rate $\mu$ and volatility $\sigma$ until a random
event that occurs with a rate $r$ resets its dynamics [28]. The reset event
can be interpreted as a worker that left the job market (for example by
retiring, being laid off or after an injury) and is substituted by another
younger worker with a starting income $x_{r}$ [27] Further details about srGBM
can be found in the Supplementary Information (SI) Section S1.

### MFPT in srGBM

How long does it take for a worker having a certain initial income $x_{0}$ to
reach a target threshold $y$ with intermittent resets to a new and fluctuating
income $x_{r}$? This time is random owing to the fact that the growth of
income is inherently stochastic and furthermore there is a temporal
stochasticity induced by resetting. The MFPT estimates the expected time for
an ensemble of such processes [13, 14, 15] of reaching income $y$ when
starting with $x_{0}$ and is given by (Section S2)

$\langle
T_{r}(x_{0},y,x_{r})\rangle=\frac{1-\tilde{T}(x_{0},y,r)}{r\,\tilde{T}(x_{r},y,r)},$
(1)

where

$\displaystyle\tilde{T}(x_{0},y,r)=\left(\frac{x_{0}}{y}\right)^{q_{1}},\quad
q_{1}$
$\displaystyle=\frac{\sqrt{(\sigma^{2}-2\mu)^{2}+8r\sigma^{2}}+(\sigma^{2}-2\mu)}{2\sigma^{2}}.$
(2)

We observe that larger growth of income (Fig. 2(a)) and larger randomness in
the economy (Fig. 2(b)) decrease the MFPT in srGBM (see also SI Section S3 for
numerical methods). These two notable factors indeed comply with typical
observations, i.e., it is easier for workers to move across the income
distribution when the economy is growing at faster rates and when there is
more volatility [29]. Furthermore, we observe that there is an optimal
resetting rate ($r^{*}$) which minimizes the MFPT $\langle T_{r}\rangle$ in
Eq. (1). In economics terms, the resetting rate can be seen as a controlled
factor that policy makers can control in order to optimize the dynamics of
income within an economy. For instance, they can develop policies that are
aimed at increasing/decreasing the number of workers who retire or leave their
job as a means to reduce time required for a worker to move across the income
distribution [30, 31, 32]. In this context, we find two additional important
facts that can help policy makers: First, for a fixed volatility, larger
growth ($\mu$) decreases the optimal rate at which the MFPT is at a minimum
level (inset of Fig. 2(a)). Policies that lead to efficient qualification
programs, or to quality foreign investments can, for example, be attributed to
this effect. Second, for a fixed growth rate, when the randomness in the
system is increased, we also observe an increased optimal resetting rate
(inset of Fig. 2(b)). This, for example can be a result of increased intrinsic
differences between societies when it comes to getting new jobs.

### Transition matrix vs. Stochastic process approach for MFPT

Differently from our approach, the state of the art methods for studying
income dynamics rely on utilizing the properties of the income transition
matrix [33]. But, as we will show here, transition matrices are in many ways
inferior to our framework.

An income transition matrix aggregates the income movements over a given
period in time [9]. It summarizes the mobility in a stochastic matrix
$\mathbf{A}$ in which the elements $A_{kl}$ quantify the probability that an
individual in income quantile $k$ in time $t$ is found in income quantile $l$
in time $t+\Delta$. Formally, let $\mathcal{S}_{k}(t)$ denote the set of
individuals that are part of quantile $k$ in time $t$. Then,

$\displaystyle
A_{kl}=\frac{|\mathcal{S}_{k}(t)\cap\mathcal{S}_{l}(t+\Delta)|}{|\mathcal{S}_{l}(t+\Delta)|},$
(3)

where $|\mathcal{S}|$ is the cardinality of $\mathcal{S}$ [8].

In a perfectly mobile economy, the entries of the transition matrix are all
equal to each other. Realistic income transition matrices, however, are
characterized with higher mobility at the bottom of quantiles than at the top.
To illustrate this, in Fig. 3(a) we display the income transition matrix for
the United States (for the period between 1989-1998) by using data taken from
Ref. [33]). In Fig. 3(b) we show that the income transition matrix in srGBM
can easily reproduce these properties by conducting a simple experiment. The
experiment is based on generating an artificial economy with a population of
one million workers. Each worker is assigned an initial income that is drawn
from the stationary srGBM distribution with parameters that best fit the real
income transition matrix (Fig. 3(a)). The dynamics of the income is then
simulated for a sufficiently long time and the mobility dynamics across a 10
year period are aggregated into deciles. The srGBM generated transition matrix
is able to explain 83% of the variations in the real transition matrix, thus
suggesting that the model indeed adequately reproduces the real world income
mobility.

Using the information contained in the transition matrix, we can quantify the
MFPT for an individual to move between two quantiles $k$ and $l$ using various
methods (see SI Section S4 and Refs. [34, 35, 36, 37]).

Before we detail the similarities and differences between the srGBM and the
transition matrix MFPT (TMFPT), several of its features need to be pointed
out. First, the TMFPT is measured in $\Delta$ periods (e.g., in the cases
described in Fig. 3(a-b), the MFPT is measured in decades), but unless there
are changes in the economic conditions, its value normalized in a different
unit (e.g., years) is independent of $\Delta$. Second, the income transition
matrix is ignorant of the growth, volatility, and resetting rates. Instead,
these parameters are bulked together in the transition probabilities. Lastly,
the majority of the methods for estimation of the TMFPT rely upon discretized
time dynamics, and thus need to be modified to capture the continuous time
dynamics [34]. Hence, every estimation based on the transition matrix can only
be an approximation for the MFPT.

The relationship between the srGBM-MFPT and the TMFPT can be shown in two
different ways. First, for TMFPT between any two quantiles, we can estimate
the starting income and the target income that generate the same srGBM-MFPT.
Intuitively, the starting income belongs to the lower quantile and the target
income to the upper quantile, but in general the position of the starting
income depends on the extent of aggregation. Second, we can expand the number
($K$) of quantiles of the transition matrix until they reach the total number
($N$) of workers in the economy (and suppose that $N$ is sufficiently large).
This is also known as the finite state to continuum approach in statistical
physics [21]. The limiting case as $N\to\infty$ and $K\to N$ results in the
srGBM-MFPT.

In general, the aggregation of real world income dynamics leads to
underestimating the MFPT (see Figs. 3(c-f), for various comparisons between
srGBM-MFPT and TMFPT). This is due to the fact that the aggregation leads to
neglecting the differences in the income distribution that appear within a
quantile. That is, income within a quantile is known to have a fractal like
structure, i.e., the income distribution within a quantile also follows a
power law [16]. This implies that not everyone within the quantile will have
the same probability to change their income status. Instead, there are intra-
quantile dependencies that govern the MFPT. The TMFPT is unable to identify
this phenomenon because the aggregation essentially removes certain parts of
the income distribution and quantifies income dynamics based on quantile
positions. In stark contrast, the srGBM-MFPT captures this distribution
dependence because it models the exact starting and the target income that a
worker wants to reach.

### Applying the srGBM MFPT to real world data

We can apply the presented theoretical framework to answer real world
questions that form the basis for developing socio-economic policies, such as:
What is the time needed for a currently minimum wage worker to spend in the
workforce in order to reach a reasonable income level? Which workers are able
to reach the highest status during their working life? How easy it is for
workers to reach certain income targets? How can we optimize the time required
for a worker to improve their status in an economy?

To answer these questions, we develop a statistical methodology for estimating
the MFPT across two points of the income distribution. Our methodology, in
general, relies on two assumptions which can be relaxed depending on the
availability of data. First, we assume that the srGBM parameters can be
discretized and are able to change each year. This reflects changes in
economic conditions and policies. Then, the srGBM parameters can be estimated
directly using data for the dynamics of the income distribution or taken from
other sources. Because these are only estimates for the srGBM parameters, they
are coupled with confidence intervals for the MFPT (See SI Section S5 and
Refs. [21, 20, 19]). Second, we assume that the resetting rate can be
approximated using data on the working age population who lost and/or left
their job within a calendar year. In what follows, we will display results for
the MFPT between percentiles of the income distribution, but we point out that
the results can be easily disaggregated into an even higher resolution.

Let us now show how we can use the data from [19] and our srGBM MFPT method to
estimate confidence intervals for the required time for US workers to move
between two arbitrary percentiles of the income distribution (See SI Section
S6 for more details about the statistical estimation). We will thereby assume
that there are no changes in the economic conditions once we estimate the
MFPT. That is, we will answer questions about the time properties of income
for each of the studied years under the assumption that the economy will
remain the same over the estimated MFPT. Thus, any changes in the MFPT over
the years can be attributed to changes in the economic policies or external
shocks to the economy.

#### What is the required time for a low income worker to reach a reasonable
income?

In Fig. 4(a) we show the estimated MFPT for a worker to leave the lowest 10%
and reach the level of income of the 50%, 75%, 90%, 95% and 99%, given that
they started at the middle position in the decile. We observe that the MFPT of
all these targets was decreasing and until the end of the 1990s, when they
reached their minimum. At the end of the 1990s these MFPTs drastically
increased and stayed at a high level until the last year of our analysis
(2015).

#### Which workers are able to reach the highest income status during their
working life?

Next, we investigate how long it would take a person to reach the income
levels of the 99 percentile, given that they start at an average position in
various quantiles of the income distribution (Fig. 4(b)).

Similar to the previous results, the amount of time needed to reach the
highest income status was at its lowest values in the end of the 1990s.
However, the MFPT values were much larger compared to the working life of an
individual. For instance, in these periods, the workers which started in the
95th percentile of the income distribution, had to work around 700 years in
1999 to reach the highest paid percentile, though with a very wide confidence
interval. This MFPT, remained at a consistently high level until the last year
of our analysis.

#### How easy it is for workers to reach certain income targets?

We can also analyze the fraction of the lowest income workers that constitute
the bottom decile which reach a certain target in the income distribution in a
time span that corresponds to the duration of a typical working life. For this
we can reverse the MFPT equation and estimate how many workers are able to
reach an MFPT within a given time span of X years (see SI Section S7 for the
mathematical details).

For example, the economic conditions in the US allowed only less than 2% of
workers coming from the lowest income level to reach the highest income status
in a span of 20 years ever since 1978. This percentage drops significantly as
we move to more recent years and to larger income targets (Fig. 4(c)). The
percentage becomes not much larger, if we set X to a larger value, such as 40
years (Fig. 4(d)).

#### How can we optimize the MFPT?

Finally, we can ask ourselves how close is the estimated MFPT to the optimal
MFPT and what can policy makers do to optimize it?

In Fig. 4(e) we plot the mean optimal resetting rate and compare it with the
empirical resetting rate (See SI Section S8 on the methods of estimation). The
inset plot compares the optimal MFPT and the one observed in reality. We find
that the empirical resetting rate is persistently above the optimal, starting
from 1988. In other words, it should be easily possible to decrease the MFPTs
of the US income distribution by developing policies that decrease the
resetting rate.

## Conclusion and Discussion

Socio-economic policies aimed at improving the welfare of individuals often
rely upon estimates for the typical time-frame required for workers to improve
their income status. Yet, state-of-the-art methods for these estimates can
only offer an aggregated view. Here, we developed a disaggregate measure by
exploiting the MFPT, a simple estimate for this time, in the srGBM baseline
model for income dynamics. We found that the srGBM-MFPT adequately reproduces
the real world features of income dynamics, often depicted in an income
transition matrix. We showed that the srGBM-MFPT is an orthogonal measure to
standard metrics. It provides a deeper insight into the time properties of
income within an economy as it is able to uncover the workers which are able
to move across the whole space of the income distribution during their working
life. This includes moves within an aggregated single level of the transition
matrix approach. Also, we created an empirical methodology for applying our
theoretical results. We showed that the methodology is equipped with a set of
tools which make it capable of answering specific questions about the time
properties of income for any economic structure, and presented its application
using US income data. Hence, we expect it to be of wide interest to
practitioners for developing policy interventions aimed at optimizing the
time. To facilitate further research and policy interventions, we share a
simplified code for calculating the srGBM-MFPT at
https://colab.research.google.com/drive/1quV4bdaNaGVmUB1EA0XOINqWC93Y8JBX.

We point out certain limitations to our approach. First, these estimates are
based on srGBM which can offer a good but nevertheless information about the
income dynamics. Our framework assumes that all workers have the same socio-
economic attributes that may affect the evolution of income, such as education
or gender, and does not differentiate across different types [38, 39, 40, 41,
41, 42, 43, 44]. Instead, it captures the attributes of workers by two
parameters: the growth of income and the volatility. We emphasize that the
methodology can be generalized and applied to more sophisticated situations by
assuming that the growth rates and volatilities are type dependent or when
they are themselves random variables (e.g., via doubly-stochastic models)
[21]. Then, each worker type will have their own MFPT (estimated using the
same equation). This may uncover even more detailed information about the time
properties of the income distribution and considerably enhance the
interpretation of our results. Indeed, this appears to be fruitful topic for
expanding the srGBM-MFPT framework.

Second, here we studied only one aspect of socio-economic inequality, the
income distribution and its evolution. The American dream is also related to
wealth inequality and its dynamics, two aspects that are yet not captured by
our framework. Indeed, recent studies have suggested that the time properties
of the realistic wealth distribution may not exhibit the features which are
usually assumed in policy practices [45, 46]. Hence, extending the framework
to account for wealth may be a non-trivial future contribution.

Third, we restricted our empirical analysis to only one economy. We do not go
into a detailed investigation on the geographical distribution of srGBM, which
might be essential for understanding the spatial differences in various socio-
economic phenomena.

Lastly, we emphasize that the application which we presented here is only
descriptive. We invite policy practitioners to delve deeper into the
interpretation of the questions arising from our empirical findings. That is:
Is the American Dream alive if it takes around 70 years for a middle-income
status US worker to reach the top 1%? Is it reasonable that the economic
conditions allow only 1% of the lowest income workers to reach the highest
income rank? What were the reasons for the sub-optimal resetting rate over the
years? In this context, the MFPT can provide much more information than the
extent of income mobility within an economy: it is able to uncover the
feasibility of each worker to move across the income distribution [47].

Yet, despite these limitations, the srGBM MFPT framework significantly
improves upon the state of the art by providing a method that is more
comprehensive, and also more accurate, at investigating the dynamics of a
worker’s income. This methodology advances our understanding of the time
dimension of an individual worker’s income, and improves the current state-of-
the-art. It will motivate new multidisciplinary research focused on creating
even more comprehensive methods that can be used to predict the time
properties of income – mean and higher order metrics – in specific domains of
the economy and be applied to economies all across the globe.

## Materials and methods

### Geometric Brownian Motion with stochastic resetting

We denote the income of a worker at time $t$ by $x(t)$. As explained in the
main text, the income dynamics empirically follows srGBM and can be described
by the following stochastic equation

$\displaystyle dx(t)$ $\displaystyle=(1-Z_{t})x(t)\left[\mu dt+\sigma
dW\right]+Z_{t}\left(x_{0}-x(t)\right),$ (4)

where $dt$ denotes the infinitesimal time increment and $dW$ is an
infinitesimal Wiener increment, which is normal-variate with $\langle
dW_{t}\rangle=0$ and $\langle dW_{t}dW_{s}\rangle=\delta(t-s)dt$. Here
$\delta(t)$ denotes the Dirac $\delta$-function. Resetting is implemented with
a random variable $Z_{t}$ which resets the income dynamics to the initial
value $x(0)=x_{r}=$ minimum observed income. To be specific, $Z_{t}$ takes the
value $1$ when there is a resetting event in the time interval between $t$ and
$t+dt$; otherwise, it remains zero. For simplicity, we assume that the
probability for a reset event, i.e., $\text{Prob}(Z_{t}=1)$ in the interval
$dt$ is given by $rdt$. In the limit $dt\to 0$, this confers to an exponential
waiting time density, namely $re^{-rt}$ for resetting events. Notice that
resetting can take the value of income $x(t)$ to go below $x_{0}$, but it
cannot make it negative.

### Empirical values of srGBM parameters in USA

We use the yearly estimates for the parameters needed to estimate the srGBM
provided in [19] (see also SI Section S5). Moreover, each year, the true
resetting rate is approximated with the fraction of the working age population
(15-64) in the USA who lost and/or left their job within a calendar year.
Workers who lost their job are those that either are temporarily laid off as
well as those who permanently lost their jobs. The job leavers, on the other
hand, are those that quit and immediately began searching for a new work. We
take these data from the dataset for unemployment provided by the U.S. Bureau
of Labor Statistics. The time series covers the period from 1977 up to 2015
and can be accessed at https://fred.stlouisfed.org.

### Estimating the srGBM MFPT in the USA

In order to study the first passage times in the US income distribution we
need to specify an initial position and an absorbing boundary. We estimate
these quantities by providing a starting percentile and a target percentile,
then we translate these values to the corresponding initial position, and
target income in the estimated income distribution, which is calculated by
running an srGBM simulation of a model economy, using the fitted empirical
values of the model parameters.

To be more specific, if we want to estimate the average time span before a
typical worker of the bottom percentile reaches a top percentile, then we fix
their threshold percentile income as a starting position of the stochastic
process (srGBM) and the target percentile “entry level” income as a virtual
absorbing boundary. The minimum observed income within the population is set
as the resetting point.

## Acknowledgments

The authors acknowledge financial support by the German Science Foundation
(DFG, Grant No. ME 1535/12-1). Arnab Pal gratefully acknowledges research
support from the Department of Science and Technology, India, SERB Start-up
Research Grant Number SRG/2022/000080 and Department of Atomic Energy, India.
Trifce Sandev was supported by the Alexander von Humboldt Foundation.

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Figure 1: Approaches to measuring the time properties of income: (a) Bar chart
for the income within a population in an initial time period. (b) Bar chart
for the income within a population in a target time period. In both (a-b) the
workers are sorted according to their income in descending order. (c)
Transition matrix $\mathbf{A}$ measuring the transition probabilities between
income quantiles for the workers in (b) if they started in (a). The entries of
the matrix describe the probabilities to move between two quantiles, and the
MFPT $\langle T\rangle$ is a function of $\mathbf{A}$. (d) In our approach,
the income dynamics of a worker is a stochastic process the MFPT $\langle
T\rangle$ to reach the target income in (b) if a worker started in (a) is a
function of parameters that quantify the state of the economy.

Figure 2: Dependence of the MFPT in srGBM on the model parameters (a) MFPT in
srGBM as a function of the resetting rate for various drift rates $\mu$. (b)
Same as (a), only for various volatilities $\sigma^{2}$. The black hollow
circles show the optimal resetting rate. The inset plots show how the optimal
resetting rate evolves as a function of $\mu$ and $\sigma^{2}$. More details
about this optimal behavior can be found in SI Section S2.

Figure 3: Transition matrix MFPT vs. srGBM MFPT. (a) Income transition matrix
in USA for the period 1989-1998, taken from [33]. (b) Income transition matrix
estimated through srGBM with parameters $\mu=0.10$ year-1, $\sigma^{2}=0.03$
year-1, $r=0.041$ year-1, chosen to match the empirical transition matrix in
(a). This was done by minimizing the Frobenius norm of the difference matrix
between the empirical and the one derived from srGBM. The srGBM transition
matrix is able to explain 83% of the variations in the original transition
matrix. (c) MFPT (in years) between deciles using the transition matrix
approach from the data in (a). (d) MFPT (in years) between deciles calculated
using the srGBM MFPT approach. (e) Difference between the MFPT obtained from
the transition matrix (c) and srGBM approaches (d). (f) Also shows the
difference between the two approaches by investigating the MFPT to a target
income as a function of the starting income.

Figure 4: MFPT in the US income distribution as a function of time. In the top
panel we show (a) The MFPT for a worker in the bottom 10% to reach 50%, 75%,
90%, 95% and 99%. (b) The MFPT for reaching 99% of the income distribution for
a worker belonging in 10%, 50%, 75%, 90% and 95%. In the bottom panel we show
the fraction of poorest 10% which reach the income of the 50%, 75%, 90%, 95%
and 99% in (c) 20 years, (d) 40 years, (e) Empirical resetting rate and mean
optimal resetting rate as a function of time. The inset plot gives the
corresponding MFPT for the optimal and the empirical resetting rate. The color
filled regions represent two standard error confidence interval bands. The
srGBM parameters are the same as those estimated in Ref. [19].

## Supplementary Information

###  S1 Properties of srGBM

In the literature, resetting is described as an approximation for the external
forces that influence the income dynamics and ensures a stationary
distribution [28, 48, 49, 50, 51]. We can use the renewal approach (see [48])
to show that that the probability density function (PDF) corresponding to
$x(t)$ has a stationary solution. In particular, the PDF with resetting
$(r>0)$ can be written as

$\displaystyle P_{r}(x,t|x_{0},x_{r})$
$\displaystyle=e^{-rt}P_{0}(x,t|x_{0})+r\int_{0}^{t}e^{-ru}P_{0}(x,u|x_{r})\,du,$
(S1)

where $P_{r/0}(x,t|x_{0}/x_{r})$ is the PDF of the reset/reset-free income
dynamics. The reset-free PDF is a log-normal function (following Itô
convention) and reads [52, 53]

$\displaystyle P_{0}(x,t|x_{0})$
$\displaystyle=\frac{1}{x\sqrt{2\pi\sigma^{2}t}}\exp\left(\frac{-\left[\log(\frac{x}{x_{0}})-(\mu-\frac{\sigma^{2}}{2})t\right]^{2}}{2\sigma^{2}t}\right).$
(S2)

Finding the PDF $P_{r}(x,t|x_{0},x_{r})$ at all times usually is a daunting
task, however a large time limit which is independent on the initial
condition, $x_{0}$, can be obtained by making use of the final value theorem
[54]. Namely,

$\displaystyle
P_{r}^{ss}(x|x_{r})=\lim_{t\to\infty}P_{r}(x,t|x_{r})=\lim_{s\to
0}s\hat{P}_{r}(x,t|x_{r})=r\hat{P}_{0}(x,r|x_{r}),$ (S3)

where $\hat{f}(s)=\mathcal{L}[f(t)]=\int_{0}^{\infty}e^{-st}f(t)\,dt$ is the
Laplace transform of the function $f(t)$. Substituting Eq. (S1) into the above
relation, it can be shown that the stationary distribution follows a power
law,

$\displaystyle
P_{r}^{ss}(x|x_{r})=\frac{r\sigma^{2}}{\alpha\sigma^{2}+\left(\mu-\frac{\sigma^{2}}{2}\right)}\left\\{\begin{array}[]{l
l l}\vskip 3.0pt plus 1.0pt minus
1.0pt&\left(\frac{x}{x_{r}}\right)^{-\alpha-1},&x>x_{r},\\\
&\left(\frac{x}{x_{r}}\right)^{\alpha+2\left(\mu-\frac{\sigma^{2}}{2}\right)-1},&x\leq
x_{r},\end{array}\right.$ (S6)

where

$\displaystyle\alpha$
$\displaystyle=\frac{-(\mu-\sigma^{2}/2)+\sqrt{(\mu-\sigma^{2}/2)^{2}+2r\sigma^{2}}}{\sigma^{2}},$
(S7)

is the shape parameter. Emergence of the power law is the fingerprint of the
real-world income distributions [55, 56]. Other stylised facts that are
recovered by the model are: larger $\mu$ (larger average population growth),
larger $\sigma$ (more randomness in the dynamics) and/or smaller $r$ (less
retiring or layoffs), result in a smaller shape parameter and a heavier-tailed
distribution. This leads to higher inequality and lower mobility in the
economy. Thus, srGBM is a minimal model that is able to adequately represent a
range of real word situations. As such it has been implemented to date in
various empirical studies (see for example [21, 19]).

###  S2 Derivation of the first passage time in srGBM

In this section, we present the derivation for the average first passage time
for the srGBM. To this end, we employ the renewal framework that allows us to
write the observables in the presence of resetting in terms of the observables
with $r=0$. Naturally, the bedrock of these studies relies upon obtaining
exact results for the resetting free process. Since we are interested in the
first passage quantities, the building blocks are usually based on the first
passage time density or the survival probability function. In what follows, we
first compute these quantities exactly for $r=0$ case, and then make the
connection to the framework to obtain the moments.

Our starting point is to write the Fokker Planck equation for the distribution
of the random income variable $x(t)$ that follows the geometric Brownian
motion with $r=0$. This reads

$\displaystyle\frac{\partial}{\partial
t}P_{0}(x,t|x_{0},t_{0})=\mathcal{L}(x_{0})P_{0}(x,t|x_{0},t_{0})~{},$ (S8)

where $\mathcal{L}(z)=\mu z\frac{\partial}{\partial
z}+\frac{\sigma^{2}}{2}z^{2}\frac{\partial^{2}}{\partial z^{2}}$ is the
generator for the GBM process (following Itô convention). Computation of the
first passage time density requires one to set the appropriate boundary
conditions. This can be done by assuming that the GBM is constrained on some
domain $\mathcal{D}$. A useful measure to compute the first passage properties
is the so-called survival probability $Q(x_{0},t)$ which can be defined in one
dimension (without any loss of generality) as follows – the probability that
the income $x(t)$ has not reached a threshold $y$ up to time $t$ starting from
a initial value $x_{0}$. Formally, this is defined as

$\displaystyle Q(x_{0},y,t)=\int_{x_{0}}^{y}dx~{}P_{0}(x,t|x_{0},t_{0})~{},$
(S9)

so that it satisfies the Fokker-Planck equation

$\displaystyle\frac{\partial}{\partial
t}Q(x_{0},y,t)=\mathcal{L}(x_{0})Q(x_{0},y,t)~{}.$ (S10)

The above equation is supplemented by the following conditions

$\displaystyle\text{Initial condition}~{}~{}Q(x_{0},y,t=0)$
$\displaystyle=1,~{}~{}x_{0}\neq y,$ (S11) $\displaystyle\text{Boundary
condition}~{}~{}Q(x_{0}=y,y,t)$ $\displaystyle=0,~{}~{}\forall t.$ (S12)

The first passage time density $f_{T}(t)$ is related to the survival
probability via [13]

$\displaystyle f_{T}(t)=-\frac{\partial}{\partial t}Q(x_{0},y,t)$ (S13)

and its moment generating function (MGF) is given by

$\displaystyle\tilde{T}(x_{0},y,s)\equiv\langle
e^{-sT}\rangle=\int_{0}^{\infty}~{}dt~{}e^{-st}~{}f_{T}(t)~{}.$ (S14)

The MGF is also related to the survival function in Laplace space quite
trivially – by taking the Laplace transform in Eq. (S13), we have

$\displaystyle\tilde{T}(x_{0},y,s)=-sq(x_{0},y,s)+1~{},$ (S15)

where $q(x_{0},y,s)=\int_{0}^{\infty}dt\,e^{-st}Q(x_{0},y,t)$ is the Laplace
transform of the survival probability. Further, we have used the initial
condition (S11). The Laplace transform then trivially satisfies [see Eq.
(S10)]

$\displaystyle\left[\mathcal{L}-s\right]q(x_{0},y,s)=-1~{},$ (S16)

with appropriate boundary conditions. Translating to the first passage
function, we arrive at the following eigenvalue equation for the moment
generating function

$\displaystyle\mathcal{L}\tilde{T}(x_{0},y,s)=s\tilde{T}(x_{0},y,s)~{},$ (S17)

We now assume that in one dimension, the threshold is at $y>0$ so that

$\displaystyle\tilde{T}(x_{0}=y,y,s)=1.$ (S18)

Moreover, near $|x_{0}|\to 0$, we should have

$\displaystyle\tilde{T}(x_{0},y,s)<\infty~{}$ (S19)

since starting from $x_{0}\to 0^{+}$, it will take a finite time for the
stochastic process to hit $y$, and thus the first passage time can not
diverge. The solution for Eq. (S17) with the above boundary conditions reads

$\displaystyle\tilde{T}(x_{0},y,s)$
$\displaystyle=Ax_{0}^{q_{1}(s)}+Bx_{0}^{q_{2}(s)},$ (S20)
$\displaystyle\text{where}\quad q_{1}(s)$
$\displaystyle=\frac{\sqrt{(\sigma^{2}-2\mu)^{2}+8s\sigma^{2}}+(\sigma^{2}-2\mu)}{2\sigma^{2}}>0$
(S21) $\displaystyle\text{and}\quad q_{2}(s)$
$\displaystyle=-\frac{\sqrt{(\sigma^{2}-2\mu)^{2}+8s\sigma^{2}}-(\sigma^{2}-2\mu)}{2\sigma^{2}}<0,$
(S22)

for any combination of $\mu$ and $\sigma^{2}$. To be consistent with the
boundary condition (S19), we should have $B=0$, and thus

$\displaystyle\tilde{T}(x_{0},y,s)=\langle e^{-sT}\rangle=Ax_{0}^{q_{1}(s)},$
(S23)

where $A$ can be calculated using the other boundary condition (S18). Putting
all the pieces together, we have

$\displaystyle\tilde{T}(x_{0},y,s)=\left(\frac{x_{0}}{y}\right)^{q_{1}(s)},\quad
x_{0}\leq y.$ (S24)

Eq. (S24) is a very useful result since it allows us to compute all the
moments for the first passage times by noting

$\displaystyle\langle
T^{n}(x_{0},y)\rangle\equiv(-1)^{n}\frac{d}{ds}\tilde{T}(x_{0},y,s)|_{s\to
0},$ (S25)

which will also be useful to obtain the first passage time moments in the
presence of resetting (see below).

The first passage quantities under resetting (i.e., $r>0$) can be related to
the underlying reset-free (i.e., $r=0$) quantities using the renewal formalism
[15, 57, 48, 58, 59]. Our starting point is to write a renewal equation for
the survival probability $Q_{r}(x_{0},y,x_{r},t)$ (similar to $Q(x_{0},y,t)$
in Eq. (S9)) which is defined as the probability that the process has started
from $x_{0}$, got reset at $x_{r}$ and stayed below the boundary coordinate
$y$ upto time $t$. The renewal equation for the survival function reads [57]

$\displaystyle
Q_{r}(x_{0},y,x_{r},t)=e^{-rt}Q(x_{0},y,t)+\int_{0}^{t}~{}d\tau~{}re^{-r\tau}~{}Q(x_{0},y,\tau)~{}Q_{r}(x_{r},y,x_{r},t-\tau).$
(S26)

Taking Laplace transform on the both sides of Eq. (S26) and after some
manipulations we arrive at the following relation

$\displaystyle q_{r}(x_{0},y,x_{r},s)$
$\displaystyle=\frac{q(x_{0},y,s+r)}{1-rq(x_{r},y,s+r)}$ (S27)
$\displaystyle=\frac{1-\tilde{T}(x_{0},y,s+r)}{s+r\tilde{T}(x_{r},y,s+r)},$
(S28)

where, while going from the first line to second, we have used Eq. (S15) i.e.,
the relation between first passage time density and survival function in the
Laplace space. The mean first passage time is then given by

$\displaystyle\langle
T_{r}(x_{0},y,x_{r})\rangle=\int_{0}^{\infty}~{}dt~{}tf_{T_{r}}(t)=q_{r}(x_{0},y,x_{r},s\to
0)=\frac{1-\tilde{T}(x_{0},y,r)}{r\tilde{T}(x_{r},y,r)},$ (S29)

where $\tilde{T}(x_{0},y,r)=\left(\frac{x_{0}}{y}\right)^{q_{1}(r)}$. This is
the central formula behind the MFPT analysis that was announced in the main
text. We have verified Eq. (S29) with numerical simulations in Fig. (2). The
results show an excellent agreement between them.

A simple observation at the Fig. (2) shows us that the MFPT can be optimized
as a function of the resetting rate $r$. This optimal rate can be computed
from the relation

$\displaystyle\frac{\partial}{\partial r}\langle
T_{r}(x_{0},y,x_{r})\rangle\bigg{|}_{r=r^{*}}=0.$ (S30)

Although it is not possible to obtain an exact expression for $r^{*}$, one can
use a numerical approach. The optimal resetting shows interesting behaviors in
terms of the intrinsic system parameters such as $\mu$ and $\sigma$ as can be
seen from Fig. 2. While the optimal rate is non-zero for small values of
$\mu$, it eventually becomes zero for large $\mu$. On the other hand, this
behavior is reversed as $\sigma$ is varied from small to large. Such a
transition of the optimal resetting rate from a finite value to zero is
reminiscent of the canonical thermodynamic phase transition that can further
be understood within the Landau like mean field theory (see [60] for a general
theory and such instances in physics).

###  S3 Numerical Estimation of MFPT through srGBM

The basic ingredient used to numerically simulate srGBM is to generate a
trajectory using Eq. (4). Concretely, to obtain the distribution of the
position of the particle at time $t$, we discretize the time $t=n\Delta t$,
where $n$ is an integer. We initialize the position of the particle at
$x(0)=1$, and then, at each step $(\tau=1,...,n)$, the particle can either
reset or it can evolve according to the laws of GBM. Thus,

  1. 1.

with probability $1-r\Delta t$ ($r$ is the rate of resetting); the particle
undergoes GBM so that

$\displaystyle x(\tau\Delta t)=x[(\tau-1)\Delta t][\mu+\sigma\sqrt{\Delta
t}\eta(\tau\Delta t)],$ (S31)

where $\eta(\tau\Delta t)$ is a Gaussian random variable with mean 0 and
variance 1, and $\Delta t$ is the microscopic time step;

  2. 2.

with complementary probability $r\Delta t$, resetting occurs such that

$\displaystyle x(\tau\Delta t)=x(0)=1.$ (S32)

The length of a single first passage event can be estimated as follows: we
simulate the stochastic process described in Eqs. (S31) and (S32) until the
particle hits a predefined virtual absorbing boundary, $x(\tau\Delta t)=y$.
The length of this first passage event is simply $\tau\Delta t$. In order to
illustrate how we can numerically extract a series of first passage time
events, we identify the sections of three sample trajectories which start at
$x(0)=1$ and end at a predetermined boundary, $x(\tau\Delta t)=y$, that is
represented by the solid line in Fig. S1. Given these first passage events, we
can easily calculate the corresponding first passage times by taking their
time span. The average of these time lengths gives an estimate for the MFPT of
the stochastic process and generally can be written as:

$\displaystyle\langle\hat{\tilde{T_{r}}}\rangle=\frac{\Sigma_{N}\,\tau\Delta
t}{N}.$ (S33)

As we increase the number of sampled trajectories, $N$, the estimated
empirical MFPT should converge to the analytical value.

Figure S1: First passage time events. Here we plot three typical srGBM
trajectories with their corresponding first passage times (circled points).
Every trajectory starts at $x_{0}=1$ (horizontal dashed line). The horizontal
bold line at $x=2$ represents the predefined target. The black vertical lines
represent a resetting event. Here, we have assumed $x_{0}=x_{r}$ without any
loss of generality.

###  S4 Estimation of the MFPT through transition matrices

An income transition matrix aggregates the income rankings and summarizes
mobility in a stochastic matrix $\mathbf{A}$ in which the elements $A_{kl}$
quantify the probability that an individual in income quantile $k$ in period
$t$ is found in income quantile $l$ in period $t+\Delta$.

Mathematically, the entries of the transition matrix can be defined as
follows. Let $\mathcal{S}_{k}(t)$ denote the set of individuals that are part
of quantile $k$ in time period $t$. Then,

$\displaystyle
A_{kl}=\frac{|\mathcal{S}_{k}(t)\cap\mathcal{S}_{l}(t+\Delta)|}{|\mathcal{S}_{l}(t+\Delta)|},$
(S34)

where $|\mathcal{S}|$ is the cardinality of $\mathcal{S}$. For example.
suppose that we want to calculate the element $A_{1,10}$ of the transition
matrix. Visually, this entry represents the probability that the highest paid
worker of the bottom quantile (see Fig. S2(a)) in time $t$ reaches the
threshold income of the lowest paid worker of the top quantile (see Fig.
S2(b)) in time $t+\Delta t$. This probability is calculated as a fraction of
the individuals that cross the minimum income of the target decile in $\Delta
t$ as shown in Fig. S2(c) and (d). We can easily generalise this procedure and
compute the full transition matrix A after arbitrary time steps.

Figure S2: Construction of a transition matrix. (a) Income distribution
histogram for the lowest paid individuals at time $t$. (b) Income distribution
histogram for the highest paid individuals at time $t+\Delta$. (c) Left
diagram: the set of individuals that belong to the bottom quantile at $t$.
Right diagram: the set of individuals that belong to the top quantile at
$t+\Delta$. (d) Transition probability from the bottom to the top quantile in
time $\Delta t$ is calculated as the fraction of individuals that reached the
threshold income of the target quantile, namely from the black colored bar in
(a) to the blue colored bar in (b).

The income transition matrix that is generated from real world data follows
approximately continuous time dynamis. The standard methods for estimating the
MFPT from a transition matrix, however, assume discrete time. Conveniently,
$\mathbf{A}$ can be easily transformed to a so called embedded Markov chain
matrix $\mathbf{\tilde{A}}$ whose MFPT estimated using discrete time methods
is the same as if we were to estimate the continuous time MFPT from
$\mathbf{A}$.

In what follows, we describe the procedure for generating $\mathbf{\tilde{A}}$
and estimating the corresponding MFPTs.

The fist step is to generate the Markov generator matrix $\mathbf{Q}$ of
$\mathbf{A}$ that describes the rates at which workers move across quantiles.
The elements $Q_{kl}$ of $\mathbf{Q}$ are

$Q_{kl}=\Bigg{\\{}\begin{array}[]{lr}\log(A_{kl}),&\text{if }k=l\\\
A_{kl}\log(A_{kl})/(1-A_{kl}),&\text{otherwise}.\end{array}$

Using $\mathbf{Q}$, $\mathbf{\tilde{A}}$ can be estimated as

$\displaystyle\mathbf{\tilde{A}}$
$\displaystyle=\mathbf{I}-\texttt{diag}\left(\mathbf{Q}\right)^{-1}\mathbf{Q},$
(S35)

where $\mathbf{I}$ is the identity matrix and
$\texttt{diag}\left(\mathbf{Q}\right)$ is a matrix with entries equal to
$Q_{kl}$ if $k=l$, and $0$ otherwise.

The MFPT between two quantiles $k$ and $l$ of a transition matrix $A$ with $K$
quantiles is related to the vector that describes the stationary transition
probabilities $\mathbf{\pi}^{{}^{\prime}}=(\pi_{1},\pi_{2},...,p_{K})$ of the
discrete Markov chain given with $\mathbf{\tilde{A}}$. The matrix $\mathbf{M}$
whose entries $M_{kl}$ give the MFPT between $k$ and $l$ can be estimated in
the following way:

  1. 1.

Compute the stationary probability vector $p^{*t}$ of the Markov chain
described with $\mathbf{\tilde{A}}$. This vector is given by the left
eigenvector corresponding to the largest eigenvalue of $\mathbf{\tilde{A}}$.

  2. 2.

Compute the fundamental matrix $\mathbf{Z}$, as
$\mathbf{Z}=[\mathbf{I}-\mathbf{A}+\mathbf{1}\mathbf{\pi}^{{}^{\prime}}]^{-1}$.

  3. 3.

Compute the MFPT matrix $\mathbf{M}$ with mean first passage time from
quantile $k$ to quantile $l$ defined as
$M_{k,l}=\frac{Z_{ll}-Z_{kl}}{\pi_{l}}$.

We refer the reader to Refs. [34, 35, 36, 37] for more details about the
estimation procedure.

###  S5 Method for empirical estimation of srGBM parameters

We assume that the income dynamics follows srGBM that is constantly under the
threat of changing its parameters. In this context, we will assume that the
resetting rate $r(t)$ is a function of time and will provide an approximation
$\hat{r}(\tau)$ with the fraction of people that lost and/or left their job.
For simplicity, we will measure the resetting rate on a yearly basis and
assume that in between two years the resetting rate is fixed, i.e.,
$r(t)\approx\hat{r}(\tau)$ for any $\tau$ between $t$ and $t+1$. Our goal is
to simultaneously provide consistent estimates $\hat{\mu}(\tau)$ and
$\hat{\sigma}(\tau)$ for the drift parameter $\mu(\tau)$ and the noise
amplitude $\sigma(\tau)$ as a function of the time, that best fits the
observed shares of income owned by the top 1% in the US income distribution.
The assumption for dynamics in the model parameters reflects the possibility
of noise in the data. In addition, it can be an approximation for the changes
in economic conditions that affect the srGBM dynamics. These can be either due
to changes in government policies or due to circumstances that are not under
the control of the policymakers.

Formally, the estimation procedure consists of the following steps:

Step 1: Fix the resetting rate $\hat{r}(0)$ in the initial period at the
initial year $\tau=0$ and then estimate $\hat{\mu}(0)$ and $\hat{\sigma}(0)$
to match the srGBM stationary distribution.

Step 2: Propagate $N$ individual income trajectories according to the laws of
srGBM. That is, with probability $1-\hat{r}(\tau)\Delta t$ the income
undergoes GBM so that:

$\displaystyle x_{i}(t+\Delta t)=x_{i}(t)+x_{i}(t)[\hat{\mu}(\tau)\Delta
t+\hat{\sigma}(\tau)\sqrt{\Delta t}\eta_{i}(\Delta t)],$ (S36)

where $\eta_{i}(\Delta t)$ is a Gaussian random variable with zero mean and
unit variance, and $\Delta t$ is a small time increment. Here, we used the Itô
convention. With complementary probability $\hat{r}(\tau)\Delta t$, the income
resets to the initial position:

$\displaystyle x_{i}(t+\Delta t)=1.$ (S37)

At last, we find the values $\hat{\mu}(\tau+1)$ and $\hat{\sigma}(\tau+1)$
that minimise the squared difference between the inferred share of the top 1%
in the modelled population in year $\tau+1$ and the observed share in real
data.

Step 3: Repeat Step 2 until the end of the time series.

For each time series we run a simulation for a model economy of $N=10^{6}$
workers. However, because of the randomness of the numeric simulations, each
simulation will result in different fitted values. To take this into
consideration, we construct a Monte Carlo estimation by repeating the process
100 times and report the average value of $\hat{\mu}(\tau)$ and
$\hat{\sigma}(\tau)$. In addition, this allows us to estimate the variability
of the results and provide confidence intervals for both parameters.

###  S6 Method for empirical estimation of srGBM MFPT

The estimation procedure for the srGBM MFPT consists of the following steps:

  1. 1.

Fix the srGBM parameters to match those estimated in [19]. Choose an initial
position and a target. These positions represent the estimated income for the
starting and target percentile, respectively.

  2. 2.

Calculate the MFPT using Eq. (1).

  3. 3.

Repeat Step 2 until the end of the time series.

###  S7 Method for estimation of Fraction of individuals reaching a target
income in X years

The estimation procedure for the fraction of individuals that reach a target
income in X consists of the following steps:

  1. 1.

Fix the srGBM parameters to match those estimated in [19]. Choose an initial
position and a target. These positions represent the average income for the
poorest individuals and target decile, respectively.

  2. 2.

Propagate $N$ individual income trajectories according to the laws of srGBM
for XX years. Calculate how many individuals $n$ out of the population $N$
reach the target income fixed in the first step.

  3. 3.

Repeat Step 2 until the end of the time series.

###  S8 Method for estimation of the optimal resetting rate

In order to estimate the optimal resetting rate for each year, we have to fix
$\hat{\mu}(\tau)$, $\hat{\sigma}(\tau)$, $x_{r}(\tau)$, $x_{0}(\tau)$, and the
target $y(\tau)$. However, given that there are multiple choices for a
starting and target position, we can estimate in total 45 different optimal
resetting rates. The minimization procedure is performed using Hessian-free
optimization, in particular the truncated Newton (TNC) algorithm.