Unraveling the dynamics of wealth inequality and the impact on social mobility and health disparities

To study changes in wealth and income in the USA through longitudinal data, we downloaded data from the Panel Study for Income Dynamics (PSID) [46], last accessed on 24 March 2023 from https://psidonline.isr.umich.edu/. We restricted our analysis to the years for which wealth information was available (i.e. years in the period 1984–2019). For these years, we also considered data related to health and food security. We complemented the study with cross-sectional data for income and wealth available from the Survey of Consumer Finance (SCV) of the Federal Reserve, considering the most recent survey (2019) last accessed on 10 March 2023 from www.federalreserve.gov/econres/scfindex.htm. Longitudinal wealth and income data for Italy was obtained from the Bank of Italy (BoI) for the years 1991–2020 ('Indagine sui bilanci delle famiglie italiane') last accessed on 22 March 2023 from www.bancaditalia.it/statistiche/basi-dati/rdc/index.html. Aggregated statistical series for food insecurity and health expenditure were obtained from the World Bank (WB) database, last accessed on 20 April 2023 at https://data.worldbank.org/. Aggregated statistical series for excess mortality were obtained from the Organization of Economic Cooperation and Development (OECD) database, last accessed on 27 April 2023 at https://stats.oecd.org/. Finally, we also consider data on the wealth of billionaires for the years 2002–2022 collected by Forbes and formatted by Gapminder www.gapminder.org/data/documentation/billionaires-dataset/ Data has been dowloaded on 18 January 2023 from https://github.com/open-numbers/ddf–gapminder–billionaires.

2.2.1. Wealth distribution

2.2.2. Wealth quantiles

2.2.3. Persistence

We study the dynamics $q_i(t)$ of the wealth quantiles associated to each household by following its time evolution for the time span available in the longitudinal surveys. For the sake of comparison, we restrict most of the analysis to a time span of 20 years. From the quantile trajectories $q_(t)$, we estimate transition probability matrices $P_Q(q^{\prime},t|q,t_0) = P(q_i(t) = q^{\prime}|q_i(t_0) = q)$ describing the probability that $q_i = q^{\prime}$ at time t when $q_i = q$ at time t₀.

The time-dependent persistence of quantile q is then defined as the probability to remain in the initial quantile

$$\begin{align} \Phi_Q \left(t|q,t_0\right) = P_Q\left(q,t|q,t_0\right). \end{align} \tag{ 1 }$$

The Shorrocks mobility index [23], often used in the economic literature to quantify social mobility, is given by

$$\begin{align} S_Q\left(t,t_0\right) = \frac{Q-\sum_{q = 0}^{Q-1}\Phi_Q \left(q,t,t_0\right)}{Q-1}. \end{align} \tag{ 2 }$$

The index ranges in $0\unicode{x2A7D} S_Q \unicode{x2A7D} 1$ and is equal to $S_Q = 0$ when the position in the wealth distribution is unchanged while complete mixing leads to $S_Q\to 1$.

2.2.4. Mean-square displacements

Persistence is an estimate of the immobility within the wealth distribution but does not inform about the patterns of mobility. To this end, we define time-dependent mean-square displacements of the quantiles as

$$\begin{align} M\left(t|t_0,q_i\left(t_0\right)\right) = \sqrt{\langle \left(q_i\left(t\right)-q_i\left(t_0\right)\right)^2 \rangle} \end{align} \tag{ 3 }$$

where the average is obtained by summing over i. If there were no restrictions to wealth mobility, in the large time limit $t \gt\gt t_0$ the position of each household within the wealth distribution is expected to be random. In this conditions, the mean-square displacement for $q_i(t_0) = q$ should approach the limit

$$\begin{align} M_\infty\left(q\right) = \sqrt{\sum_{q^{\prime} = 0}^{Q-1}\left(q^{\prime}-q\right)^2/Q} = \sqrt{\left(q^2 - q \left(Q - 1\right) + Q^2/3 - Q/2 + 1/6\right)}. \end{align} \tag{ 4 }$$

We denote this limit as 'perfect mix'.

2.2.5. Conditional expectation values

We simulate the AGM of wealth inequality proposed by Bouchaud and Mézard [30]. The model describes the evolution of the wealth W_i of the agent i, within a population of N agents, according to the following stochastic differential equations

$$\begin{align} \frac{\mathrm dW_i}{\mathrm dt} = \eta_i\left(t\right) W_i + \sum_j \left(J_{ij} W_j -J_{ji} W_i\right) \end{align} \tag{ 5 }$$

where J_ij controls wealth redistribution between agent i and agent j and $\eta_i(t)$ is a Gaussian random noise with mean m and variance σ. The multiplicative noise term $\eta_i(t) W_i$ is interpreted in the Stratonovic sense. We consider a redistribution interaction embedded in regular random network with degree c = 4 with $J_{ij} = J_0/c A^0_{ij}$, where $A^0_{ij}$ is the adjacency matrix of the network ($A^0_{ij} = 1$ if the node i is connected to j and zero otherwise) [42, 43].

To integrate numerically the equations, it is convenient to express them in terms of an N-dimensional system of equations for the vector $X = (W_0, \ldots, W_{N})$ which obeys

$$\begin{align} \mathrm dX_i = \sum_j A_{ij} X_j \mathrm dt + B_i X_i \mathrm d\psi_i \end{align} \tag{ 6 }$$

where $A_{ij} = A^0_{ij}J_0/c + \delta_{ij} (m-J_0/c\sum_k A^0_{ki} -\phi)$, $B_i = \sigma$ and $\mathrm d\psi_i$ are independent Wiener processes (with variance 1 and zero mean).

We exploit the scaling invariance of the equations [30] and study the model as a function of the reduced parameter $J_0/\sigma^2$, measuring time in units of $1/\sigma^2$. Since we are only interested in relative inequality, we rescale the wealth W_i by the total wealth $W_\mathrm T = \sum_i^N W_i$ defining the wealth fraction $\tilde{w}_i = W_i/W_\mathrm T = N w_i$ belonging to agent i [30]. In this way, the results are independent on the parameter m.

Wealth inequality is conventionally measured considering the wealth distribution or more often by defining simplex indices, such as the Gini index [21]. The upper tail of the wealth probability density has received wide attention in the literature and is well described by the Pareto distribution $P(W) \sim 1/W^{\alpha+1}$, where the Pareto exponent α ranges between 1 and 2 for most countries [37]. A lower Pareto exponent correspond to higher wealth inequality.

Here we use national survey data (PSID for USA and BoI for Italy, as discussed in the Methods section) to estimate the wealth distribution for households in USA and Italy in different years. Since the distribution spans several order of magnitudes in wealth, we analyze it using logarithmically-spaced bins, which is appropriate for a power laws. As discussed in the methods section, we consider normalized wealth since we are interested in inequalities rather than in the absolute values of wealth. The estimated wealth probability density functions, for positive wealth only, are for different years are reported in figures 1(a) and (b) for Italy and USA, respectively. As expected, we observe a Pareto tail for high wealth, with exponents $\alpha\simeq 1.45$ for USA and $\alpha\simeq 1.95$. The lower part of the distribution reveals another power law exponent W^−β , with $\beta \simeq 0.5$ which is more apparent in US data, but also observable in Italian data. We also notice that the general shape of the distribution remains the same across the years with only small deviations, so that distributions for different years are superimposed.

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We have confirmed the double power law behavior observed in PSID using an independent cross-sectional survey, the SCF by the Federal Reserve. The results shown in figure 2 confirms this result. In particular, we fit the data with the interpolating function

$$\begin{align} f\left(w\right) = C_1 \frac{w^{-\beta}}{1+\left(w/w_c\right)^{\left(1+\alpha-\beta\right)}}, \end{align} \tag{ 7 }$$

obtaining α = 1.46 and β = 0.47 as best fit parameters. The crossover value is estimated to be at $w_c = 0.87$ which corresponds to 730 000 USD.

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We now turn our attention to the part of the wealth distribution that is not included in figures 1(a) and (b): the region of zero or negative net wealth. In figures 1(c) and (d), we report the fraction of households with zero or negative wealth as a function of time for USA and Italy, respectively. The data show a much higher fraction in USA with respect to Italy and in both cases the fraction has increased in the past 30 years, reaching up to 20% in the USA. Hence, while the relative distribution of positive wealth remains roughly constant, the increase in the fraction of households that have no wealth is consistent with the general growth of wealth inequality reported in the literature.

Having established the general form of the wealth distribution, we now concentrate on the mobility of individual households within the distribution. To this end, we label households according to their wealth decile and follow their time evolution for a time span of 20 years using longitudinal national surveys for USA (PSID) and Italy (BoI). This is illustrated in figures 3(a) and (b) for USA and Italy, respectively, using a heatmap showing the time-dependent transition probability $P_Q(q^{\prime},t|q,t_0)$ that an household falling into a decile q (here $q = 0,3,5,7,9$) at $t = t_0$ ($t_0 = 1999$ for USA and $t_0 = 2000$ for Italy) is found in decile qʹ at time t. The figure show some degree of diffusion across the deciles which is, however, minimal for households starting from the upper decile.

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A summary of the mobility pattern can be obtained by computing the Shorrocks mobility index [23] S₁₀ , as discussed in the methods section. The result reported in figure 3(c) reveal little differences between USA and Italy and display a slow time dependence. The Shorrocks index, however, averages together the mobility of all the wealth quantiles into a single number and therefore does not allow to appreciate the difference in mobility of households of different wealth. To this end, we consider the time-dependent persistence into the top and bottom decile for households in the USA and in Italy as shown in figures 3(d) and (e). The results show in both cases, a marked difference in persistence between the top and bottom decile with households in the top decile being the most persistent. All the curves are well fit with a stretched exponential function

$$\begin{align} \Phi_{Q}\left(t|q,t_0\right) = \left(1-\Phi_{Q}^{\infty}\right)\exp\left(-\left(t/\tau\left(q\right)\right)^{\gamma\left(q\right)}\right)+\Phi_{Q}^{\infty}, \end{align} \tag{ 8 }$$

where $\Phi_{Q}^{\infty}$ is the value expected assuming that after an infinite time the wealth is uncorrelated with its initial value (i.e. $\Phi_{Q}^{\infty} = 1/Q = 1/10$). The best fits yield characteristic times $\tau(0) = 1.7$ and $\tau(9) = 10^5$ years with $\gamma(0) = 0.37$ and $\gamma(9) = 0.07$ for the USA and $\tau(0) = 3$ and $\tau(9) = 415$ years with $\gamma(0) = 0.27$ and $\gamma(9) = 0.13$ for Italy. Figure 3(f) reports residual persistence, defined as the value of the persistence observed after 20 years for households in different deciles minus the expected asymtpotic value (i.e. $\Phi_{Q}(t = 20)-\Phi_{Q}^{\infty}$). These results suggest that persistence depend on the initial decile and is highest for top wealth owners that are for a large part immobile. Residual persistence is instead lowest for households in intermediate wealth deciles.

Measures of persistence, including the Shorrocks index, provide information about the extent of wealth immobility of different strata in the population but do not inform about wealth mobility patterns encoded in the transition probability matrix. To this end, we compute mean-square quantile displacements $M(t|t_0,q)$ as defined in the Methods section. Figures 4(a) and (b) shows that $M(t|t_0,q)$ grows slowly in a q dependent manner, for the USA and Italy. In both cases, the slowest growth is observed for the highest quantiles the fastest for the lower one, particularly in the USA. As in the case of persistence, we can compare the observation with the result expected asymptotically if we assume that the final wealth decile is independent on the initial one (perfect mix). In this case, $M(t|t_0,q) \to M_\infty(q)$ can be computed analytically (see methods section). A comparison between the observed residual mean-square quantile displacements after 20 years and $M_\infty(q)$ is reported in figures 4(c) and (d) for the USA and Italy. The figure show that for all value of q the observed mobility patterns are far from perfect mix since mobility is rather localized. The lowest level of mobility is observed for highest wealth deciles. Furthermore, the typical changes in wealth decile over a span of 20 years is between 2 and 3.

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For completeness, we repeated our analysis of persistence and mobility using income instead as wealth. A summary of the results are reported in figure 5 for Italy and figure 6 for the USA. The results are qualitatively similar to the analysis based on wealth, except for the fact that there is less disparity in persistence and mobility between the top and bottom decile. For all cases, however, residual persistence is considerable and mobility is restricted.

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Our analysis shows that a fraction of the population in USA and Italy has either zero or negative wealth, lying at the bottom of the wealth distribution (figure 1). Moreover, studying the mobility across the wealth distribution, we observed persistence within each wealth decile that is largest at the extremes (figure 4). We then examine the possible consequences for health of persistent lack of wealth. Since PSID contains longitudinal information on how health-related variables associated with wealth for US households but we could not access comparable data for Italy, we decided to compare the two countries using aggregated statistical data.

Aggregated data show that the rate of avoidable mortality, both preventable and treatable, in the USA was almost twice as large as in Italy for the years 2010–2020 (see figure 7). It is interesting to point out that this occurs despite the fact that the healthcare expense per capita, governmental and private, is considerably larger in the USA than in Italy (figure 7).

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We then studied in detail the excess mortality due to various diseases from chronic diseases to mental ones, considering the years lost per 100 000 people for the same period (figure 8). For all the diseases considered, including diseases related to metabolism and nutrition, the circulatory system and the nervous system, mortality in the USA is larger than in Italy. The only exception is found in the case of cancer (figure 9). In fact, for colon cancer mortality in USA is higher than in Italy, but for bladder, skin and stomach cancer Italy is higher than USA. Finally, for lung, leukemia and pancreatic cancer the mortality is similar between the two countries. The complex etiology of this diseases and the possible link with prevention/therapeutic strategies that are different between the two countries as well as in general the different lifestyle (from the quality of the food to the practise of physical activities) could be involved in the different evolution of the tumors.

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Considering that in general USA show higher mortality for a broad range of diseases (the only exception being cancer), we then turn our attention on the possible association between health and persistence at the bottom of the wealth distribution, analysing the longitudinal national surveys for USA (PSID). As shown in figure 10(a), approximately 40% of the households with no wealth in the year 1999 persist in this condition until 2019. We thus consider households that had zero or negative wealth in 1999 and in 2019 (persisters) and compare these households with those that had zero or negative wealth in 1999 but a positive wealth in 2019 (non persisters), and also consider the general population (all) for reference.

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For households in the persisters group the fraction of family heads with poor health status is larger than that of family heads of non persister households which on their turn is larger that of the general population (figure 10(b)). We then look at the incidence of several conditions including cancer (figure 10(c)), diabetes (figure 10(d)), hearth disease (figure 10(e)), hypertension (figure 10(f)) and psychological problems (figure 10(g)). In particular for hearth disease, hypertension and psycological problems, the data indicates that lack of wealth and no-wealth persistence are associated with poorer health status.

Food insecurity is known to be connected to diseases such as hypertension, depression and anxiety [19]. Figure 11(a) reports the percentage of population affected by moderate or severe food insecurity in USA and Italy according to the World Bank, while figure 11(b) reports the percentage related to severe food insecurity for the period 2005–2020. We observe that food insecurity is generally larger in the USA than in Italy, but severe food insecurity is larger in Italy and on the rise in the past few years.

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We thus consider the possible relationship between food insecurity and no-wealth persistence analysing the longitudinal national surveys for USA (PSID). We follow the same strategy as for the case of health status: we fist compare aggregated data for the two countries and then analyze more closely the case of the USA. Figure 11(c) shows the fraction of households with food insecurity as a function of time subdivided into persisters, non persisters and all. We see that the percentage of households with food insecurity among persisters is not decreasing in time, being almost twice that of non persisters and three times larger than that of the general population. On the other hand, the percentage of households with food insecurity decreases among non-persisters approaching the levels of the general population after 20 years. Similar trends are observed in the mean food expenditures (figure 11(d)): persisters spend in average less than non persisters and even less than the general population.

To better interpret our findings on wealth persistence and mobility within a population, we resort to an agent based model in which a series of identical agents accumulate and loose wealth subject to randomness and redistribution. In particular, we concentrate on the model of wealth condensation proposed by Bouchaud and Mezard [30] and studied in detail in a series of subsequent papers by various authors [31, 42, 43]. The steady-state properties of the model are well understood analytically [30, 31, 42, 43] and in good agreement with empirical statistics. In particular, the model is attractive because it is able to recover a distribution of wealth which displays Pareto (power law) tails resulting from the stochastic dynamics of interacting agents in a growing economy. Furthermore, the distribution of relative wealth converges to a steady state and the Pareto exponent is controlled only by the ratio of two parameters, representing the redistribution rate (J₀) and the variance of the noise distribution (σ²).

Here we perform numerical simulations of the model focusing on its out-of-equilibrium behavior (see the Methods section for more technical details on the model). We first verify that our simulations recover known steady-state properties of the model [30], including the power law wealth distribution, the Pareto tail exponent dependence on $J_0/\sigma^2$, the system-size dependence of the inverse participation ratio and wealth condensation for low $J_0/\sigma^2$ (see figure 12).

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Figure 13(a) report the temporal evolution of the wealth quantiles (for Q = 10) of N agents as a function of the ratio $J_0/\sigma^2$ which controls the extent of wealth inequality (the lower $J_0/\sigma^2$ the higher the inequality). The figure shows that after a sufficiently long time the quantile of each agent appears to be uncorrelated with the initial value. The time to approach to this steady-state depends on $J_0/\sigma^2$, which is consistent with analytical results for the correlation time obtained in the mean-field version of the model [30]. To be more quantitative, we compute the persistence in the top and bottom decile for different values of $J_0/\sigma^2$ (see figures 13(b) and (c)) in analogy with the analysis of longitudinal surveys reported in figure 3. We thus fit the simulated persistence curves with equation (8) and obtain a convincing match for all values of $J_0/\sigma^2$ We find that the stretched-exponential exponent $\gamma(q) \simeq 0.5$ for q = 0 and $\gamma(q) \simeq 0.6$ for q = 9, with little dependence on $J_0/\sigma^2$.

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The relaxation time of the persistence $\tau(q)$ is found to depend on $J_0/\sigma^2$ so that a more unequal wealth (figure 13(d)) distribution is reflected by a longer relaxation time. Furthermore, the relaxation of the bottom decile is faster than that of the top decile. Both observations are in qualitative agreement with the data from US and Italian surveys: wealth inequality is larger in the USA than in Italy and so are the relaxation times and the relaxation time for q = 0 is larger than the one for q = 9 in both countries. We next consider the Shorrocks mobility index and see that it converges to S₁₀ with in a time that increases as $J_0/\sigma^2$ decreases (figure 13(e)). Finally, we measure mean-square quantile displacements which are illustrated in figure 13(f) for $J_0/\sigma^2 = 1$. All the curves approach a q dependent steady-state limit corresponding to the perfect mix limit $M_\infty(q)$ (equation (4)).

The simulation results discussed above is in qualitative agreement with the time evolution of wealth observed in panel surveys from USA and Italy. In particular, persistence and mean-square quantile displacements follow similar time-dependent functions, including a stretched exponential relaxation. On a quantitative level, however, there are fundamental differences between model predictions and empirical data. First of all, the large differences in persistence between households in the top and bottom decile is not explained by the model. Data show that persistence in the top decile decays extremely slowly with a large fraction of households that are effectively immobile. Furthermore, the exponent of the stretched exponential $\gamma(q)$ is smaller in the data than in the model, indicating again that that mobility is much slower in the data than what would be suggested by the model. This is corroborated by the analysis of mean-square quantile displacements that are more localized in the data than in the model. Finally, agents with zero or negative wealth are by construction not present in the model.

To understand why the wealth mobility patterns predicted by the model are not in line with the data, we reconsider the main assumption of the model, namely that the time-dependent changes of the agents wealth are directly dependent on their current wealth. This fact is summarized in figure 14(a) showing that the standard deviation of the changes in wealth conditioned to to the current wealth is proportional to the wealth itself $\sigma(\mathrm dW|W) = CW$, with C < 1. When we measure the same quantity in empirical data for USA (figure 14(b)) and Italy (figure 14(c)), we find that proportionality is only found for large wealth, while for low wealth $\sigma(\mathrm dW|W)$ becomes independent on wealth.

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To corroborate this conclusion, we analyze the evolution of personal wealth using the list of ultra-wealthy individuals compiled by Forbes for the years 2002–2022. As shown in figure 14(d), the standard deviation of wealth changes for billionaires is proportional to wealth as assumed in the model. We thus analyze persistence in the group of the 300 top wealthiest individuals. The rank dynamics is illustrated in figure 14(e) and the time-dependent persistence is reported in figure 14(f). The curve is well fitted by equation (8), setting $\Phi^{\infty} = 0$, and obtaining γ = 0.7 and τ = 13 years. Within this restricted group, rank turnover is relatively fast and well described by the agent based model.

The linear dependence of the wealth fluctuations on wealth is associated to the way personal income is generated. As shown in figure 15(a) for the case of US households, income is on average proportional to wealth for large wealth while for low wealth it is not. By analyzing the main source of income of each households, we see that for low wealth income is mainly due to wages, pension or transfers, while for higher wealth it is due to business ownership and capital gains/dividends (see figure 15(b)). The situation is similar for Italian households (figures 15(c) and (d)), although the transition is less sharp than in the USA.

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