Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

We assume that time is continuous, and there is a continuum of workers. The income $x(t)$ of an individual worker in period $t$ follows a geometric Brownian motion with stochastic resetting (srGBM). That is, it grows multiplicatively with a rate $\mu$ and volatility $\sigma$ until a random event resets its dynamics [19]. The reset event can be interpreted as a worker that left the job market (for example by retiring or being laid off) randomly with a rate $r$ and is substituted by another younger worker with a starting income $x_{0}=1$. The dynamics of the income can be described by the following Langevin equation

$\displaystyle dx(t)$

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

(1)

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. Moreover, $Z_{t}$ is a random variable which resets the income dynamics to the initial value $x_{0}$. $Z_{t}$ takes the value $1$ when there is a resetting event in the time interval between $t$ and $t+dt$; otherwise, it is zero.

The solution to Eq. (1) can be found by interpreting the srGBM as a renewal process: each resetting event renews the income dynamics at $x_{0}$, and between two such consecutive renewal events the income of an individual undergoes simple GBM. Thus, between time points $0$ and $t$, only the last resetting event, occurring at the point

$\displaystyle t_{l}(t)=\max_{k\in\left[0,t\right]}k:\{Z_{k}=1\},$

(2)

is relevant, and the solution to Eq. (1) reads (following Itô interpretation) [20, 21, 22]

$\displaystyle x(t)$

$\displaystyle=x_{0}~{}e^{(\mu-\frac{\sigma^{2}}{2})\left[t-t_{l}(t)\right]+\sigma\left[W(t)-W(t_{l}(t)\right]}.$

(3)

The probability for a reset event is given by $P(Z_{t}=1)=rdt$. In the limit when $dt\to 0$, this corresponds to an exponential resetting time density $f_{r}(t)=re^{-rt}$, and $t_{l}$ is distributed according to

$\displaystyle f(t_{l}|t)=\delta(t_{l})e^{-rt}+re^{-r(t-t_{l})},$

(4)

such that $\int_{0}^{t}~{}dt_{l}f(t_{l}|t)=1$. Intuitively, the first term on the RHS corresponds to the scenario when there is no resetting event up to time $t$ while the second one accounts for multiple resetting events.

In the literature, resetting is described as an approximation for the external forces that influence the income dynamics and ensure a stationary distribution [6, 7, 10, 23]. One can use the renewal approach as described above (also see [7]) to show that that the probability density function (PDF) has a stationary solution. In particular, the PDF with resetting $(r>0)$ can be written as

$\displaystyle P_{r}(x,t|x_{0})$

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

(5)

where $P_{0}(x,t|x_{0})$ is the PDF of the reset-free ($r=0$) income dynamics [24, 25]

$\displaystyle P_{0}(x,t|x_{0})$

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

(6)

By Laplace transform, $\mathcal{L}[f(t)]=\int_{0}^{\infty}e^{-st}f(t)\,dt=\hat{f}(s)$, of Eq. (5), and by using the limit $s\rightarrow 0$, we find the steady state, $P_{r}^{ss}(x|x_{0})=\lim_{t\to\infty}P_{r}(x,t|x_{0})=\lim_{s\to 0}s\hat{P}_{r}(x,t|x_{0})=r\hat{P}_{0}(x,r|x_{0})$. Following this, it can be shown that the stationary distribution follows the power law

$\displaystyle P_{r}^{ss}(x|x_{0})=\frac{r\sigma^{2}}{\alpha\sigma^{2}+\left(\mu-\frac{\sigma^{2}}{2}\right)}\left\{\begin{array}[]{l l l}\vskip 3.0pt plus 1.0pt minus 1.0pt&\left(\frac{x}{x_{0}}\right)^{-\alpha-1},&x>x_{0},\\ &\left(\frac{x}{x_{0}}\right)^{\alpha+2\left(\mu-\frac{\sigma^{2}}{2}\right)-1},&x\leq x_{0},\end{array}\right.$

(9)

where

$\displaystyle\alpha$

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

(10)

is the shape parameter. The power law property is an important stylised fact that is prevalent in real-world income distributions [26, 27]. 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), 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 [5]).

The behaviour of the inequality and mobility measures are crucially determined by the moments and the log moments of srGBM. Therefore, we begin the analysis by providing a short overview of their properties.

srGBM is characterised by three physical long time non-ergodic regimes that affect the dynamics of the per capita income in relation to the typical, median income in the population [16]. They are: i) a frozen state regime, ii) an unstable annealed regime, and iii) a stable annealed regime. The regimes appear because the per capita income is a “plutocratic” measure, i.e., the individuals with larger income also play a larger role in its value and therefore in certain cases the mean income may fail to depict the behaviour of the typical individual in the population. Indeed, the regime which is observed in reality depends on the relation between the resetting rate and the other parameters of the model.

Let us now turn our attention to one of the main topics of this study and examine how income inequality measures behave in the different srGBM regimes. Formally, income inequality can be seen as a static concept. It quantifies the extent of concentration in the income distribution. In this context, an inequality measure is a function that ascribes a value to the observed distribution of income in a way that allows direct and objective comparisons across different economies and points in time [31, 32, 33]. In situations when the income is concentrated on a small number of individuals the measure will suggest larger inequality, and vice versa, when the income is more spread across the population it will suggest a more egalitarian society. In the economics literature, a variety of income inequality measures have been developed (for a review, see Ref. [34]). In what follows we will analyse the underlying features of three most widely used empirical measures by assuming srGBM dynamics: the Gini coefficient, the share of income owned by the top X% individuals in the population and the Theil inequality index.

Measures of economic mobility quantify how income ranks of individuals change over time. Intuitively, when mobility is high, the chances of an individual to change their position in the income distribution over a given time period are high. In contrast, when mobility is low, individuals are unlikely to change their rank in the distribution over time, or the changes may happen slowly. As a means to study the drivers of economic mobility in srGBM we consider two standard measures: Spearman’s rank correlation and the earnings elasticity (EE). In fact, both the rank correlation and the EE are measures of immobility, and to consider them as measures of mobility one has to consider their complement or their inverse.

Quantifying the behaviour of the various inequality and mobility measures in srGBM allows us to study the relationship between them in an economy. A standard approach for visualising the relationship is through the Great Gatsby curve. The curve was introduced by Corak [3] who utilised country level data to illustrate the association between the Gini coefficient and the Earnings Elasticity, indicating that economic immobility and inequality are positively related, implying that economies with a more unequal distribution of income also offer fewer perspectives for individuals to change their income rank.

In Fig. 4 we display numerical results for the Great Gatsby curve by simulating the srGBM in the stationary state $10^{3}$ times and afterwards quantifying the boxplots for the distribution of the generated Gini coefficients and Earnings Elasticity for various resetting rates. The figure demonstrates that the model adequately reproduces the qualitative empirical observation that inequality and immobility are positively related. However, the regime dependence of the inequality measures is mirrored in the qualitative behaviour of the Great Gatsby curve. While in the stable regime, it is easy to notice the direct relationship between the Gini coefficient and the Earnings Elasticity, the direction of the relationship becomes unclear in the unstable regime. This is because then inequality is a highly volatile phenomenon and is uniquely determined by the degree of randomness in the system. In the frozen regime, the Gini coefficient reaches its maximum value, and then the direct relationship between inequality and immobility vanishes.

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 $r(t)$ to be estimated is a function of time and will provide an approximation $\hat{r}(T)$ 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., $r(t)\approx\hat{r}(T)$ for any $t$ between $T$ and $T+1$. Our goal is to simultaneously provide consistent estimates $\hat{\mu}(T)$ and $\hat{\sigma}(T)$ for the drift parameter $\mu(T)$ and the noise amplitude $\sigma(T)$ as a function of the time, that best fits the observed Shares of income owned by the top 1% and the top 10% 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 $T=0$ 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 $1-\hat{r}(T)\Delta t$ the income undergoes GBM so that:

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

(21)

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

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

(22)

At last, we find the values $\hat{\mu}(T+1)$ and $\hat{\sigma}(T+1)$ that minimise the squared difference between the inferred share of the top 1% (or top 10%) in the modelled population in year $T+1$ 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 $\hat{\mu}(T)$ and $\hat{\sigma}(T)$. In addition, this allows us to estimate the variability of the results and provide confidence intervals for both parameters.

Each year, the true resetting rate $r(t)$, is approximated with the fraction of the working age population (15-64) in USA who lost and/or left their job within a calendar year. The people 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.

The data for the top 1% and top 10% used for assessing the robustness were taken from the The World Inequality Database (https://wid.world/).

The empirical results are shown in Fig. 5. The figure displays the inferred regions of the regimes based on the fitted values $\hat{\mu}(T)$ and $\hat{\sigma}(T)$, together with the approximation of the resetting rate as a function of time for the share of the top 1% (Fig. 5(a)) and the share of the top 10% (Fig. 5(b)). The inset plots provide the fitted and the shares observed in reality. It can be seen that in both cases, the fitted share accounts for more than $95\%$ of the observed cross period variation, (the value of the coefficient of determination $R^{2}$ in each case is above $0.95$), and therefore it can be argued that the model adequately predicts the income dynamics in the United States. More importantly, when we look at the evolution of the regimes in the country, we find that the fitted growth rate of income is persistently above the resetting rate $\hat{r}(T)$ (black line) in both inference procedures. This leads us to the conclusion that the process is robustly in the frozen regime (the highlighted blue region) for the time period considered. Only in the period after the Financial Crisis of 2010, when modelling the top 10% we observe a change towards the unstable regime (orange region), although this change is relatively small. Hence, the evidence we present here suggests that the US income dynamics is in a state where it is uniquely determined by the frozen regime and thus the dynamics is characterised by non-ergodicity.