Inequality, mobility and the financial accumulation process: a computational economic analysis
Abstract
Our computational economic analysis investigates the relationship between inequality, mobility and the financial accumulation process. Extending the baseline model by Levy et al., we characterise the economic process through stylised return structures generating alternative evolutions of income and wealth through time. First, we explore the limited heuristic contribution of one and twofactors models comprising one single stock (capital wealth) and one single flow factor (labour) as pure drivers of income and wealth generation and allocation over time. Second, we introduce heuristic modes of taxation in line with the baseline approach. Our computational economic analysis corroborates that the financial accumulation process featuring compound returns plays a significant role as source of inequality, while institutional arrangements including taxation play a significant role in framing and shaping the aggregate economic process that evolves over socioeconomic space and time.
Keywords
Inequality Economic process Compound interest Simple interest Taxation Computational economicsJEL Classification
C46 C63 D31 E02 E21 E27 D63 H221 Introduction and literature review
Capital wealth accumulation is an evergreen matter of economic analysis and policy.^{1} Public finance and financial macroeconomic models define notions of production, income and capital wealth and study their aggregated evolution over time (Bertola et al. 2006; Blanchard 2011; Snowdon and Vane 2005), as well as their distribution across individuals (see the survey of economic literature by Sahota 1978). In particular, some notable efforts aim to explain empirical distributions as driven by either certain stochastic processes, or the combined influence of driving factors such as family environment, talent, education and social status. Recent economic modelling strategies include Nirei and Souma (2007) and Gabaix et al. (2016).
Since the fifties, the standard representation of growth denotes a multiplicative process that is also the standard representation for individual financial investments. This modeling strategy implicitly assumes a single capital stock that is measured and reinvested for the aggregate economy over time (Perroux 1949; Stone 1986). Recent advances in dynamic macroeconomic modelling, based upon the representative agent hypothesis, have been criticized for disregarding the aggregate dimension featured by collective and dynamic phenomena (Gallegati and Kirman 1999). Previously neglected, issues of income and wealth distributions have gained socioeconomic momentum in the aftermath of the Global Financial Crisis of 2007–2008, including through the 99% movement in US (Haldane et al. 2014; Alvaredo et al. 2013). This movement claims that the increased financialisation of economy and society involves an increased appropriation of income and wealth by the richest 1% of the population at detriment of the remaining 99%, leading to more unequal and allegedly unfair distributions of income and wealth. This distributional issue renewed theoretical interest through influential positions taken by leading economists (Krugman 2013, 2014a, b; Stiglitz 2012; Solow 2014) and policymakers (Haldane 2014)), as well as through the publication of economic history studies conducted by Thomas Piketty and Emmanuel Saez among others, reconstructing longrun statistical time series of income and wealth distributions in US and abroad (Atkinson et al. 2011; Piketty 2014; Piketty and Saez 2014).
According to Haldane et al. (2014), “as ever, dispute rages about the precise statistics. But the longterm patterns are clear enough–and remarkable. Almost half of the growth in US national income between 1975 and 2007 accrued to the top 1% (OECD 2014). In the UK and US, the top 1%’s share of the income pie has more than doubled since 1980 to around 15% and their share of the wealth pie has been estimated at up to a third–more than the whole bottom half of the population put together (ONS 2013; Wolff 2012). The five richest households in the UK have greater wealth than the bottom fifth of the population (Oxfam 2014)”.^{2}
The theoretical issue of income and wealth distributions is wellknown since classic economic theorists in the XIX century at least, when the leading economist Mill (1861) considered “fair and reasonable that the general policy of the State should favour the diffusion rather than the concentration of wealth.” At the beginning of the XX century, the leading economist and sociologist Pareto (1895, 1965, 1897a, b) argued for the socalled Pareto (powerlaw) wealth distribution as an empirical regularity, while the economic statistician Gini (1912) developed ingenious statistical measurement techniques to capture this inequality through the socalled Gini index.^{3}
Recent advances in econophysics point to the functional forms of statistical distributions of income and wealth (Lux 2005). In particular, some scholars aim to reproduce empirical regularities through simple, elegant additive economic processes (Angle 2006; Richmond and Solomon 2001; Solomon and Richmond 2002). Other scholars purport to explain the fat tail of these distributions (that is, the tail concerned with the higher ranges of aggregate income and wealth) through multiplicative economic processes which lead to emerging powerlaws (Levy 2005; Levy and Levy 2003; Milakovic 2003). Some recent contributions suggest the form of a deformed exponential function, which seems to capture well the empirical regularities of income distribution at the lowmiddle range, as well as its powerlaw tail (Kaniadakis 2001, 2002). These modelling attempts have raised a lively debate with some economists who were worrying about allegedly poor socioeconomic understanding and lack of theoretical economic underpinnings (Gallegati et al. 2006; Lux 2005). Further collaborative and interdisciplinary research has developed the application of the kdeformed exponential function to the parametric modelling of personal income and wealth distributions (Clementi and Gallegati 2015, 2017; Clementi et al. 2007, 2008, 2009; Clementi et al 2010; Clementi et al. 2012a, b, 2016). The latter approach provides insights on the drivers of these distributions over time and across the population, while enabling synthetic comparison through inequality and poverty measures that are derived from parametric estimations.
In this context, generalising Champernowne (1953), Levy (2005) and Levy and Levy (2003) (Levy et al. thereafter) have developed an elegant modelling strategy purporting to explain the powerlaw tail of income and wealth distributions under financial market efficiency, and the stochastic distribution of financial returns across individuals active in this market.
 (i)
the inequality of income and wealth allocation across individuals, and its evolution over time;
 (ii)
the significance of collective institutional mechanisms, including taxation, that actively frame and shape this economic process.
The rest of the article is organised as follows. The second section introduces a financial accumulation process model inspired by the Levy et al. model, as baseline scenario. The third section shows the implications of this model for the evolution of inequality and social mobility through time, assessing their sensitivity to changes in variance and nonnormal distribution of returns. The fourth section extends the baseline model by introducing decreasing returns and the simple return structure. The comparison with this latter structure corroborates that, without financial accumulation, inequality is not increased over time in the baseline scenario. The fifth section introduces a second flow factor (labour income) along with the stock factor (capital wealth) considered by Levy et al. The introduction of a flow factor may involve an incomesaving process that complements and integrates the financial accumulation process driven by inherited wealth. All together, the analysis developed in the first five sections makes clear that distributional effects, which depend on aggregate configurations, have been neglected by the received literature. This preliminary conclusion paves the way to introducing minimal institutions (à la Shubik) that denote collective mechanisms related to income and wealth distributions. In particular, the sixth section introduces simple centralised modes of taxation, featuring a proportional taxation model (proportional taxation of periodic net income, uniformly redistributed through provision of universal public service), and a progressive taxation model (progressive taxation of periodic net income, redistributed in a regressive way through direct transfers). A summary of main results concludes.
2 Modelling strategy for the financial accumulation process
This stylised model does not pretend to reproduce economic reality in its totality. In particular, it does not introduce consumption, overlapping generations, or windfall gains and losses due to wars or accidents. However, it captures one featuring element of the aggregate economic process: financial accumulation opportunities. Compound returns feature financial investment dynamics and related institutions. Financial institutions, such as investment funds, and widespread measures of financial performance are based upon compound return as reference logic. It seems then particularly significant to disentangle and analyse its impact. The aggregate economic process is increasingly managed through corporate forms that live indefinitely and can then go on performing financial accumulation. On the one hand, financial investment is conducted by institutional investors which are driven by, and assessed against, compound return. On the other hand, eventual redistribution of their financial proceeds is often received by corporate recipients that go on reinvesting those proceeds over time, in a selfreferential financial accumulation dynamics.
Throughout all our computational analysis, we assume an initial equal distribution of wealth \(W_{i,t=1} = 10 \,\, \forall \,\, i\) across all individuals at initial time \(t=1\). This implies that inequality depends entirely on the specifications of the economic process. Furthermore, for sake of simulation, we impose the same random seed to all the various sets of simulations proposed in this article. When not mentioned otherwise, we also define a population of \(N=5000\), and we run every simulation round for \(t_{max}=5000\) steps.^{5} Contrary to Levy et al., we allow the theoretical possibility that individual wealth falls to, and remains at zero level. Individual agents take financial investment risk and may occasionally lose all their capital wealth.^{6}
Our computational economic analysis disentangles two featuring dimensions to be analysed: wealth inequality across individuals, and social mobility relative to wealth dimension.
3 The baseline case
The dynamics of wealth concentration across individuals over time is impressive under the baseline scenario introduced by Levy et al. Wealth distributions become increasingly skewed under various compound return structures where individual returns are extracted from normal and gamma distributions at each period of time. For all these structures, the upper tail of wealth distribution goes on appropriating an increasing share of aggregate wealth over time. This dynamic effect has implications for wealth inequality (Fig. 1). In particular, the Gini Index shows that wealth inequality is magnified under the baseline case, asymptotically tending to its maximal value of one. Drawing upon Fernholz and Fernholz (2014)’ proof, we introduce the following Lemma 1 concerned with the evolution of timeaverage wealth distribution:
Lemma 1
The asymptotic value of a Gini Index based upon timeaverage wealth tends almost surely to its maximum value of one.
Proof
See Appendix 7.1.
Following Biondi and Olla (2018), we further introduce the following lemma concerned with the Gini Index across the population at each point of time:
Lemma 2
The asymptotic value of the Gini Index \(G_t\) on the entire population at a certain point of time t asymptotically tends almost surely to its maximum value of one.
Proof
See Proof of Lemma 3.1 in Biondi and Olla (2018).
Our result on wealth mobility is further reinforced observing the mobility of the 1% richest individuals at different points of time. This experiment is visualized in Fig. 3. Having ranked all the agents according to their wealth at one period t, the top 1% is selected. Ranks are normalized by dividing values for the total number of agents so to establish a measure that is independent of population size. In the Left panel (Fig. 3), we compute the average position of each selected agent (that is, those included in the top 1% at period t) over the following 1000 periods. We then show the distribution of probability of this average rank position.^{7} The farther is the time period t at which the selection of the top \(1\%\) is made, the lower is the average rank position for those top individuals. This implies a decreasing downward mobility and increasing persistence of the top 1% over time. Moreover, Fig. 3 (Right panel) shows the evolution of the average rank of individuals in the top 1% at time t over the next 1000 periods. As time passes, richest individuals at a given period t tend to remain among the richest. Indeed, while individuals that are among the richest at \(t = 10, 100\) may yet revert to some lower position over time (being replaced by previously poorer individuals), individuals that are rich at \(t = 1000, 2000\) tend to remain in the top decile of the wealth distribution. Finally, individuals that are rich at \(t = 3000, 4000\) tend to remain in the top centile of wealth over time, thus perpetuating their social position relative to wealth.
The sensitivity to return variance in the baseline case is magnified by financial market dynamics. Empirical evidence for financial markets behaviour shows that actual market price and return series are not normally distributed, featuring fat tails and extreme events (Mandelbrot 1963, 1967; Mantegna and Stanley 1996; Biondi and Righi 2016, 2017, providing further references). For sake of simulation, we complement normal distribution of returns with a gamma distribution having the same mean as in the baseline case, but featuring extreme events, i.e., gamma(a, b) with \(a=0.25\) and \(b=0.2\) where \(a \cdot b=\mu _r=0.05\) and \(a \cdot b^2=\sigma _r=0.01\). This parameterization for the gamma distribution is applied to all our simulation analyses, when not stated otherwise. Computational results show that the gamma distribution of returns reinforces wealth concentration and inequality, while undermining wealth mobility over time. In particular, the Gini Index is always superior at each period of time (Fig. 2, Left panel), while the Wealth Mobility Index is always inferior (Fig. 2, Right panel). This result foreshadows that the nonnormality of financial market returns may have an inequalityenhancing impact, favouring skewed accumulation of wealth across individuals and over time.
4 Decreasing compound returns and simple return structures: History matters
Levy et al. insist on the stochastic nature of their financial investment process. Our computational economic analysis further points to its cumulative nature over time, depending on the peculiar deployment of compound returns. Along with stochastic extraction of the actual return r for investor i at each time t, the financial investment process is further featured by the cumulative impact of the individual series of compound returns on accumulated wealth through historical time (Fig. 1). A quick glance at the deterministic reduction of the process model in Eq. 2 shows that being richer at time t almost assures becoming richer at a further time \(t + n\) with \(n \gg 0\). Coeteris paribus, this evolutionary structure tends to favour investors that become richer earlier in time, that is, investors that accumulate net gains before (and net losses after) the others, since every gain compounds positively, while every loss compounds negatively through time. This cumulative process is exacerbated by the constant mean return to wealth which was assumed in the baseline scenario. Therefore, rather than ‘being lucky’, this process denotes that ‘history matters’. This financial accumulation process has important implications for the evolution of wealth through socioeconomic space and time. In a similar vein, Keynes (1933) would “trace the beginnings of British foreign investment to the treasure which Drake stole from Spain in 1580”, reinvested at annual compound return of 3.25% over the next centuries to 1930, while remembering its connection to “avarice and usury and precaution that must be our gods for a little longer still [... to] lead us out of the tunnel of economic necessity into daylight”. Hysteresis and path dependency play an important role in explaining the inequality generated by the financial accumulation process. Wealth concentration is everincreasing over time (Fig. 1), while relative social mobility is undermined by increasing differences in total wealth,^{8} as showed by the Wealth Mobility Index (Fig. 2). In sum, idiosyncratic compound returns on investment through time prove to have cumulative effects, making the aggregate distribution of wealth not stationary. In particular, this distribution becomes increasingly rightskewed over time, tending to a limit in which wealth is concentrated entirely at the top (see also Fernholz and Fernholz 2014; Biondi and Olla 2018). This rightskewed tail of wealth distribution depends especially on the second order of return distributions at a certain time period t (that is, the variance \(\sigma _r\) in case of returns that are normally distributed).
Lemma 3
The Gini Index \(G_t\) tends asymptotically to 0 for \(t \rightarrow \infty \), under simple return structure, implying perfect equality among individuals.
Therefore, wealth inequality proves to be crucially dependent on the accumulation of returns through time, while this accumulation further undermines social mobility as expressed by relative levels of wealth. In sum, the economic process modelled by Levy et al. denotes a significant connection between inequality and the financial accumulation process. This accumulation through time proves to be conducive to increased wealth inequality and decreased wealth mobility over historical time. However, its assumption of constant average returns to aggregate wealth under compound return structure seems unsustainable, because indefinite compounding cannot realistically hold in the longrun (Voinov and Farley 2007; Biondi 2011). Moreover, IMF studies (Berg and Osrty 2013; Ostry et al. 2014) show that inequality affects growth, with higher inequality being associated with lower growth. It seems then unrealistic to assume a structurally stable economic process while the upper wealth tail goes on appropriating an everincreasing part of total wealth, with the Gini Index asymptotically reaching the maximum value of one in the longrun. However, our model clearly shows the logical and institutional tensions between the multiplicative logic embedded in compound return, and this potential tradeoff between inequality and growth. Let alone, such a multiplicative process may involve a selffulfilling decrease in returns, reducing then social welfare.
In order to extend this onefactor model, the next section shall introduce a twofactors model of the economic process, adding a flow factor which features an additive evolution to the stock factor characterised by a cumulative evolution through time.
5 A model of aggregate economic process combining capital wealth (stock) and labour (flow) factors
According to our configuration, the wealth return \(r_{i,t}\) defines the wealth yield \(Y^W_{i,t} = r_{i,t}W_{i,t1}\). The fair condition imposes that this wealth income is equal, on average, to the saved income of the period. The latter is defined as \(Y^L_{i,t} = s_{i,t} Y^L_{t}\).
 (i)
all saving rates are equally likely, and
 (ii)
the average saving rate is always equal for all the individuals.
According to our computational results (Fig. 8), even this progressive reinvestment of savings cannot reshape wealth inequality and mobility under the fair condition stated above. Wealth distributions and all the indexes maintain the same behaviour as under the baseline case.^{9}
Although each individual draws savings from labour income according to a uniform distribution, these savings are reinvested according to the same financial accumulation process as the inherited wealth, which is initially equal for all agents. Therefore, savings from labour income do not introduce an alternative reference logic or a complementary economic process. Consequently, they cannot reshape wealth distribution, inequality and mobility over time.
In sum, under both the Levy et al. model and the widespread twofactors model of the aggregate economic process, wealth concentration, inequality and mobility depend crucially on the compound return structure that characterises the accumulation of financial investment over time. The decomposition of wealth dynamics in two factors of productions does not change its evolution over time.

The entire notion of ‘factor of production’ is an incubus on economic analysis, and should be eliminated from economic discussion as summarily as possible.

I have to insist again that anyone who reads my 1955 article [Solow (1955)] will see that I invoke the formal conditions for rigorous aggregation not in the hope that they would be applicable [. . .] but rather to suggest the hopelessness of any formal justification of an aggregate production function in capital and labor.
6 The impact of taxation and redistribution
Taxation is a classic matter related to income and wealth distributions. Available statistics do extensively rely upon fiscal data for gathering evidence (Saez and Zucman 2014; Topritzhofer et al. 1970). It is then interesting to explore its effect on wealth concentration, inequality and mobility by expanding the baseline model by Levy et al. through featured modes of taxation.

Concerning tax levy, we assume either a uniform proportional tax rate for all individuals (uniform taxation), or a progressive tax rate increasing with the tax base (progressive taxation).

Concerning tax distribution, we assume either a uniform redistribution to all individuals (featuring the provision of universal public service), or a regressive redistribution that decreases with the tax base (featuring provision of direct transfers for welfare to the polity).
Finally, under progressive taxation and welfare model (Progressive and Welfare), the tax authority employs the tax levy denoted by Eq. 19 in order to redistribute the total levied amount in a regressive proportion to wealth, according to Eq. 18. This model features the provision of direct transfers to the polity, funded by progressive tax levy.
Stylised modes of taxation and redistribution
Tax Redistribution  Proportional  Regressive 

Tax Levy  Redistribution Rate  Redistribution Rate 
Proportional tax rate  Proportional taxation and public service  Proportional taxation and welfare 
Progressive tax rate  Progressive taxation and public service  Progressive taxation and welfare 
The actual relative impact on wealth inequality and mobility across modes of taxation depends on the parameter space assumptions. Computational results (Fig. 10) for mean tax and redistribution rates explain why progressive taxation is less effective than proportional taxation in reshaping wealth inequality and mobility, under our framework of analysis. As mentioned above, individual tax rates under progressive tax regimes depend on the underlying distribution of wealth. Since the latter is increasingly rightskewed over time, this dependency involves a mean tax rate that progressively becomes and remains very low over time (materially inferior to the mean tax rate of 0.05 applied under proportional tax regimes). Consequently, the impact of progressive tax regimes over wealth inequality is materially reduced both in absolute terms, and relative to proportional tax regimes which apply an exogenous fixed tax rate. Moreover, since wealth distribution is increasingly and materially rightskewed over time, a relatively low tax rate is sufficient to asymptotically stabilise the Gini Index (Fig. 9, Left panel). Wealth is so concentrated on the top (see Lemma 1) that a relatively low tax extraction from the richer is sufficient to materially increase wealth of the poorer, involving a stabilising effect on wealth inequality over time (Fig. 10, Bottom panel). This result does not establish preference for, or superiority of proportional tax regimes over progressive tax regimes. In particular, the effectiveness of the proportional tax system, while guaranteeing higher tax collection, neglects the regressive nature of this tax mechanism. A policy implication of this result is that effectiveness of fiscal systems depends on the underlying economic structure and process. Therefore, our analysis would recommend tax authorities committed to progressive tax regimes to maintain tax rate structures based on absolute wealth thresholds and independent from relative wealth levels. The latter tax authorities should secure a sufficient degree of progressiveness of taxation, as well as a sufficiently high top tax rates.
In conclusion, taxation materially reduces wealth concentration and inequality, compensating the impact of financial accumulation process. Taxation proves therefore to be effective in counterbalancing the inequality effects of the financial accumulation process. This result is consistent with Fernholz and Fernholz (2014) arguing that “the presence of redistributive mechanisms then ensures the stability of the distribution of wealth over time”.
7 Concluding remarks
The poet Trilussa mocked national statistics to be that accounting method for which, one individual having eaten two chickens and another one just none, both would result to have eaten one chicken each.^{10} Students of income and wealth distributions may keep this adage in mind while developing related macroeconomic models, especially under the representative agent assumption.
Footnotes
 1.
Hereafter, the term “capital wealth” combines concepts of capital and wealth to stress the productive nature of wealth considered by our economic analysis. Indeed, we especially point to financial investments, while durable assets held for consumption are excluded from our analysis.
 2.
See also CBO (2011).
 3.
Literature on the Pareto (powerlaw) distribution of wealth is too vast to be summarized here and outside the purpose and scope of this article, which is not concerned with the statistical form of wealth distribution. Further readings include: Kirman (1987), Dagum (1990), Drgulescu and Yakovenko (2001) and Persky (1992).
 4.
In fact, Levy (2005) (chapter, p. 111, footnote 13) concedes that even joint accumulation processes with heterogeneous accumulation of talents are asymptotically Paretian, with the fasterincreasing multiplicative process dominating the highrange in the long run. Fernholz and Fernholz (2014) maintain that, in their model, “luck alone  in the form of high realised investment returns  [...] creates divergent levels of wealth.”
 5.
The Matlab code of the simulations can be found at: https://github.com/simonerighi/BiondiRighi2018_JEIC.
 6.
For simulation purpose, we calibrate the parameter space to make this possibility unlikely. In the scenarios presented in this article, despite the high number of iterations, no agent ever loses its wealth completely. Agents experience partial losses (negative returns), but no complete loss of their wealth.
 7.
The time average of rank positions reduces the impact of idiosyncratic oscillations. Similar results are obtained by replacing the average rank positions with the individual positions after 1000 periods.
 8.
Levy and Levy (2003, p. 7) prove that “the actual wealth distribution converges to a Pareto distribution \(\left[ \dots \right] \) with minimum wealth, average wealth, and variance that grow over time” when a lower bound on minimum wealth is introduced. Without the latter, the distribution converges to a nonstationary lognormal distribution.
 9.
The same results hold when actual returns are derived from simple return structure. Computational results are available under request.
 10.
“Me spiego: da li conti che se fanno \(\backslash \) seconno le statistiche d’adesso \(\backslash \) risurta che te tocca un pollo all’anno: \(\backslash \) e, se nun entra nelle spese tue, \(\backslash \) t’entra ne la statistica lo stesso \(\backslash \) perchè c’è un antro che ne magna due.” (Trilussa, La statistica).
Notes
Acknowledgements
We dedicate this article to the memory of Prof. Pierpaolo Giannoccolo, coauthor and dear friend of us. Pierpaolo had been a committed team member and was working with us on furthering the understanding of the financial economic process. We thank Alan Kirman, Thomas Piketty, Stefano Olla, Shyam Sunder and Marco Valente for their insightful comments and suggestions. Previous versions of this article were presented at the ‘International Economic Law and the Challenge of Global Inequality’ Conference, King’s College London (17–8 April 2015), the 20th Annual Workshop on the Economic Science with Heterogeneous Interacting Agents (WEHIA 2015), the Applied Economics Lunch Seminar, Paris School of Economics (2 June 2015), the Workshop on “Financial Markets and Nonlinear Dynamics” (FMND, Paris, 4–5 June 2015), and at the Econophysics Colloquium (Prague, 14 September 2015).
Supplementary material
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