Simple wealth distribution model causing inequality-induced crisis without external shocks
SSimple wealth distribution model causing inequality-induced crisis without externalshocks
Henri Benisty
Laboratoire Charles Fabry, Institut dOptique Graduate School, CNRS,Univ. Paris Saclay, 2 Ave Augustin Fresnel, 91127 Palaiseau Cedex, France (Dated: July 8, 2018)We address the issue of the dynamics of wealth accumulation and economic crisis triggered byextreme inequality, attempting to stick to most possibly intrinsic assumptions. Our general frame-work is that of pure or modified multiplicative processes, basically geometric Brownian motions.In contrast with the usual approach of injecting into such stochastic agent models either specific,idiosyncratic internal nonlinear interaction patterns, or macroscopic disruptive features, we proposea dynamic inequality model where the attainment of a sizable fraction of the total wealth by veryfew agents induces a crisis regime with strong intermittency, the explicit coupling between the rich-est and the rest being a mere normalization mechanism, hence with minimal extrinsic assumptions.The model thus harnesses the recognized lack of ergodicity of geometric Brownian motions. It alsoprovides a statistical intuition to the consequences of Thomas Piketty’s recent “ r > g ” (returnrate > growth rate) paradigmatic analysis of very-long-term wealth trends. We suggest that the“water-divide” of wealth flow may define effective classes, making an objective entry point to cali-brate the model. Consistently, we check that a tax mechanism associated to a few percent relativebias on elementary daily transactions is able to slow or stop the build-up of large wealth. Whenextreme fluctuations are tamed down to a stationary regime with sizable but steadier inequalities, itshould still offer opportunities to study the dynamics of crisis and the inner effective classes inducedthrough external or internal factors. PACS numbers: 42.79
I. INTRODUCTION
Mathematical models for economy using microscopicagent-based descriptions have attracted a lot of attentionin the last few decades [1–12]. They draw on the richtools of physics to describe some characteristic observedtrends in several complex fields. Notably, various featuresof the statistical distribution of wealth among individualsor entities (firms, cities, etc.), especially those featuringpower-law distributions (Pareto tails or Zipf’s law), havebeen studied within assumptions of simple stochastic in-gredients [13–17].Furthermore, nowadays, the degree of inequality inwealth distribution as well as its evolution are issues ofgrowing interest. A witness of this worldwide interest,beside the echo of extreme wealth inequality as yearlyreported by Oxfam for instance, is the success of ThomasPiketty’s analysis [18, 19], namely the “ r > g ” paradigm(where r is the return rate of capital and g the growthrate of the whole economy): from a “law” that is de-ceivingly simple, historical analysis of long-time series ofpatrimonial wealth and incomes across centuries suggeststhat its implications at the multi-decadal scale are pos-sibly very large.Adverse or beneficial consequences of inequality inagent-based models are mostly thought in terms of someexplicit extra variable(s) with threshold or similar pro-cedures that amount, from a physicist’s point of view,to nonlinearity. The economics narrative translates thisin various ways, within current political biases [20]: The “trickle down” effect suggests that any “added value”created by the large means of the affluent shall, sooneror later, diffuse down all social strata of society and incuran overall benefit. Features that can be explicitly consid-ered are for instance the advent of monopolies and howtheir adverse effects on competition, pricing, innovation,firm creation and death can be tracked. Note also thata majority of wealth distribution studies stick to a staticview, even if fine rendering of actual data is sought, in-cluding for instance the role of inheritance and bequeststrategy [21].The generic idea that such nonlinearities or extra para-metric bias then induce crisis, and change the growthregime from smooth to moderately or highly intermit-tent, has been acknowledged generically in the broadwake of John Maynard Keynes, and specifically by theeconomist Hyman Minsky. More recently, the workby the “heterodox” economist Steve Keen could sub-stantiate recent trends in escaping the “representativeagent model” and its questionable ability to describecrisis mechanisms. In statistical econophysics mod-els, the “wealth condensation”, that describes the ad-vent of extreme events, extreme inequality in particu-lar [3, 9, 18, 22], emerged in the first years of the disci-pline.However, the lure of describing accurately Pareto tailsor other fat tails epitomizing inequality led to the dy-namics of wealth distribution being rarely investigateduntil a few years ago. The input of econophysics hingeda lot on equilibrium thermodynamics, with its ability to a r X i v : . [ q -f i n . E C ] A p r bridge micro to macro and predict emergences such asphase transitions for instance. So, non equilibrium ther-modynamics, that is, how wealth distributions adapt topermanently moving landscapes and possibly never relaxto a steady state, was rather left aside.A compounded effect on this state of affair is the factthat the Geometric Brownian Motion (GBM) of “agent”ensembles, a tenet of stochastic studies of wealth distri-butions, leads, when left to evolve freely, to nonstationarydistributions: essentially log-normal distributions withdrift and broadening width over time. The lack of ergod-icity of these ensemble distributions was recently pointedout (i.e. the average wealth of a population at time t doesnot converge at all, at large times, to the same valueas the average of individual asymptotic fates). This ismathematically no more than an issue of noncommutinglimits [23]. But the lack of recognition of this issue hasplausibly induced several weaknesses in mainstream eco-nomics, as recently pointed out [24, 25], the most strikingbeing the outright rejection of diverging utility functionsin models. Studying the heart of the topic, conversely,led to a possibly more intrinsic metric of inequality basedon the logarithm [24, 25].Practically, it is of course desirable to confront the“socially agnostic” GBM distributions to a descriptionof growing inequalities [26]. This can be done by at-tempting to track the large inequality modulations in thelast century through a reallocation mechanism acting asa restoring force [27]. Doing so, it appears that, evenwithout any social bias such as lower education of thepoor class or the likes, the best description of the lastfour decades is one of diverging inequality and negativereallocation. The chiasm is large compared to a con-ventional “adiabatic” picture of a stationary distributionthat would evolve close to a local equilibrium with gen-tle disturbances from economic factors addressed throughvarious “output gaps” [27].The relevance of models in relation with their dynam-ics rather than their equilibrium distribution is also ad-dressed by Ref. [26], as the study finds that any bareGBM would cause too slow a dynamic, once a plausi-ble stationary calibration of the GBM is done. Onlyby introducing a nonlinearity related to an inner soci-ety structure, can the dynamics be appropriately fitted.The work by Liu and Serota [28] shows that nontrivialnonstationary mathematical distributions can be char-acterized through the time-constants and correlations oftheir momenta. So the issue of modelling (and calibrat-ing) both the transverse (population) and longitudinal(time-dependent) wealth distributions is becoming theminimally significant scope to exploit such models andget insight from them.Along this scope, it seems logical to explore all thedynamics of GBM-based models, and notably involvingtheir most striking feature, related to non ergodicity, theinequality “condensation”, whereby the wealthiest indi- FIG. 1. (Color online) (a) Gaussian distribution of w − w p atincreasing times as indicated (log-scale), with β = 0 .
06 andinitial distribution concentrated at w ; (b) Wealth distribu-tion P ( w ) on a linear scale. viduals not only capture a large fraction of the total,but also generate the largest positive or negative fluctua-tions. For instance, there is no clear intuition whether afew time-constants can be relevant to describe realisticsdynamics, or if a wilder evolution, with a “noisy” spec-trum is the better picture. It is therefore interesting toexploit the background of GBMs to explore such issues,with the view that they can reveal subsets and dynamicsthat are not an obvious part of the conventional wisdomin the way the science of complex systems is applied toeconomics.In this paper, we focus in the spirit exposed above on amodel whose intermittence is related to high inequality,but whose nonlinearity is as implicit as possible. We thusavoid to cast a moral stance on microscopic behaviorsand their intended modifications. Still, we believe thatthe way it reveals the dynamics of inequality suggests avision of the diagnoses to be made in actual economiesand of the possible counter-measures to be associated.Such visions could help broadening the much demandedalternative points of view to the so-called conventionalwisdom.In a nutshell, we examine a model whereby, accordingto the oxymoronic say, “the tail is wagging the dog”, i.e.,the presence of a tail of a few very large wealth moves thedistribution as a whole. Its discrete nature can be antic-ipated to blur simplified collective dynamics, e.g. thosewith simple time constants. The interest of this emphasisis to attract the attention to the general features (in timeand in instantaneous distribution) that are likely to linkexcessive inequality and a situation of uninterrupted cri-sis reminiscent of the last decades of worldwide economictroubles. Generality is granted here by simplicity, nottaking the bias of a “mean-field” approach with an aver-age representative agent, but rather outlining the “space-time” roles of the extremes [29]. Further mapping on var-ious topologies of networks [10, 30] could of course helpunderstanding, as well as a cross-analysis with emergingGBM dynamics studies [21, 26, 27]. Our exploitation willbe to identify a “water divide” of wealth flow, openingopportunities to view the separated “basins” as classes.The basis of our model is a simple set of N randommultiplicative processes that describe the daily fate ofagents wealth [1, 2, 5–11, 17, 31]. Needless to say, multi-plicative process are associated to interest rates, but asis usual for GBMs in this context, we do not model any-thing but “wealth”, and have no time horizon (i.e. nolong term correlation, finite agent life, etc.) in micro-scopic features. If we denote w j ( t ) generic variables atinteger times t (days) submitted to a multiplicative pro-cess, described by a probability distribution Π( λ ) dλ toget w j ( t + 1) = λw j ( t ), we have an additive process byconsidering their logarithm x j = log( w j ). We initiallystick to the balanced case where the average gain ex-pectancy of agents is zero, that is (cid:82) λ Π( λ ) dλ = 1 [15, 16].This makes more clear the dynamic role of extremes, asthe primary evolution without sizable extremes in thedistribution is a gentle unstructured noise, compatiblewith the impression of a fair game.In Section II, based on the known log-normal distribu-tion, we develop what happens in this fully independentbut yet discretized evolution [15]: We give a few “cali-bration” hints to justify a rather high daily “bet”, inde-pendent of the Kelly criterion, noting the unsatisfactorystatus of time constant calibration [27]. We explain whathappens when starting from an idealized Dirac-type egal-itarian wealth distribution P ( w ) (much as in [23, 25, 27]accounts). We use the underlying log-normal distribu-tion of x j , which has a residual nonzero drift because (cid:82) (cid:96)π ( (cid:96) ) d(cid:96) < (cid:96) = log( λ ) describes the additiveprocess deriving from the multiplicative one [1, 2, 14–16].The evolution of this distribution causes a strong inter-mittency regime occurring when the wealthiest agentsreach a large fraction of total wealth [3, 10]. The driftcomponent results, at long times, in a global impover-ishment of all agents. We assess the role of log( N ) indetermining essential time constants. Note that we im-pose a nonzero (“floor”) lower wealth. Although it hasno influence in this Sec.II, it is consistent with the nextsections and affects the post-condensation fate.In Section III, we introduce the simple feedback mech-anism of normalizing the average wealth to its initial value [5, 9, 13, 15, 29, 30]. We show that this results inendless intermittency that affects the whole wealth spec-trum. Here, our “wealth floor” plays a dynamical role, asit provides a flux even after a large collapse. We confirmthe picture that “the tail is wagging the dog”, i.e., thatthe wealthiest agent fluctuations are those that impacton the rest, by studying the correlation of ordered wealthseries. There are two aspects here: first, we observe howthe dynamics can be momentarily much shorter than thelog( N ) one. Secondly, we identify a “water divide” ofwealth flow. There is therefore an effective closure be-tween system size (thus, N ), system collective dynamics(crisis and intermittency), and this inner divide.In Section IV, we deduce how a damping mechanismcould act, and summarize the possible meaning and use ofthe results in Sec.V. If our diagnosis holds, then prevent-ing the build-up of too large entities should be averted.A brief “stylization” discussion is made [18–20, 22, 30].Our choice in Section IV is to bias the daily “exchange”,which has zero-change expectation in the non-dampedmodel [ (cid:82) λ Π( λ ) dλ = 1]. We do so by favoring a smallgain of the poorest, and a small loss of the richest, byan amount which is very small (0.5-2%) compared tothe daily “bet”. This choice was inspired by the pric-ing mechanisms introduced by Aristotle, whereby theprice has no absolute underlying reference that the mar-ket should “discover”, but is rather related to the “socialstatus” of the agent, as Paul Jorion pointed out from hisanthropological studies of various communities [32]. Thisview does not really contradict the usual pricing law ofsupply and demand in a linear regime (a continuum ofstatus), but it allows to extend it to extreme cases, en-abling a survival revenue notably, i.e., forms of solidaritythat go beyond mere greed and are present in an “em-bedded” view of economy, to use Karl Polanyi’s words.We show that small amounts of this bias are effective insuppressing the intermittency and result in a stationaryself-replicating wealth distribution. Then the equilibriumdistribution can be found as the solution of a tractableeigenvalue problem. The eigenvalue spectrum could alsohelp further dynamical studies. Indeed, a remaining de-gree of nonstationarity could also be part of the requiredstylization of human economies. Let us finally underlinethat we have no substantial consideration for the wealthsdistribution in terms of Pareto tail or power law [15], let-ting this for further work, as obviously there is interestin the topic [17]. II. THE N-AGENT MULTIPLICATIVE MODELIN THE CASE OF MERE RANDOM WALK
We consider here N agents of wealth w j ( t ) at discretetimes t = 1 , , . . . To study the wealth distribution intrin-sically, we set the average wealth at the start as a con-stant w . We want to account for the exchange of wealthand information among agents without any explicit mi-croscopic mechanism. The simplest assumption is thateducation of agents teaches them to stay on the crest ofgain and loss on the average. We then have a zero-sumexchange, with only individual fluctuations [2, 5, 6, 9].We thus model the process as a multiplicative one (oftencalled Gibrat’s law) with some simple added features de-tailed below: at each time (each day) the agent engages agiven fraction β of his wealth. To make the model easierto grasp in terms of metaphor, we set it up as follows: w = 1000 (1) w p = 400 (2) w j ( t + 1) = w p + λ ( w j ( t ) − w p ) (3)where we use for the multiplier λ a rectangular distribu-tion uniformly spanning [1 − β, β ] for simplicity:Π( λ ) = Π rect (cid:18) λ − β (cid:19) = 12 β rect (cid:18) λ − β (cid:19) (4)where rect is unity in the interval (cid:2) − , (cid:3) . Clearly, (cid:82) ∞ λ Π( λ ) dλ = 1 , i.e. there is no ensemble averagechange in wealth in an individual process. The readershould nevertheless be aware of the nonergodicity of suchensembles [23]. We will see later the time scales at whichcare is needed, at least when initial conditions are canon-ical. Eq.2 introduces w p = 400 as a minimum “floor”wealth. It is important to set the ratio w p w to valuessuch as 0.4 here that at least grossly represent devel-oped economies. A simple aspect is that in this way, wewill stick to an underlying Brownian motion (a simpleBrownian motion for the logarithm of w ) and preserve adecent value for total wealth at the slumps, without needto introduce a barrier or other nonlinearity at the smallwealth end of the distribution. Also, when it later comesto the feedback (Sec.III, Sec.IV), the coupling itself willhave noticeable effects because it acts on nonzero wealthfor the (many) poorest agents, hence a likely role on dy-namics. Thus, while it introduces an unneeded featurein the present section, we retain this poverty “floor” hereand throughout the paper.As is well known [1, 2, 5, 9, 13, 15, 23, 25, 31], start-ing from a given initial state at t = 0, we have a diffu-sion+drift process in the x = log( w − w p ) space. Startingfrom an initially single-valued distribution P ( w, t = 0) ≡ δ ( w − w ), in other words a distribution concentrated at x = log( w − w p ), the distribution of the variable w − w p undergoes two evolutions : it spreads diffusively like aGaussian in x space, thus taking the form of a parabolain a log-log representation, and its center also drifts.Both effects are determined by the second and firstmomenta of the distribution of (cid:96) = log( λ ), denoted π ( λ ),which obeys π ( (cid:96) ) d(cid:96) = Π( λ ) dλ . It is found by standardalgebra that in our case of zero-average exchange of Eq.4, the drift velocity (per unit time in x space) is given by: ν drift = 12 β ((1 + β ) log(1 + β ) − (1 − β ) log(1 − β )) − (cid:39) − β β . The dis-tribution and its standard deviation σ ( t ) in x -space takethe form:ˆ P ( x, t ) = 12 π √ t exp (cid:18) − ( x − ( x + ν drift t )) σ ( t ) (cid:19) (6) σ ( t ) = 2 β (cid:114) t π (7)as depicted in Fig.1a in log scale of w and Fig.1b in lin-ear scale of w . The fact that there is a drift for a zero-exchange distribution is one of the several not-so-intuitiveaspects of GBMs (see the example of Fig.2 in [24] pin-pointing the less-intuitive aspect of multiplicative vs. ad-ditive processes, as appears from the authors account ofa large swath of scientific history through two Bernoullifamily members and Laplace).Apart from the two wealth ingredients w and w p , wehave to discuss a realistic value for β . If we had to dis-cuss about agents faced to optimizing their utility whenrisking their wealth in some bet, we could recourse to theKelly criterion, for instance. But our scope is distinctlydifferent: we rather want to calibrate the randomness ofthe economy as a whole (and not as a stationary distribu-tion, but rather vs. its dynamics), and thus in our spirit,the choice of β rather has to be dictated by the spreadof fate for a bunch of agents of any given initial wealth:the relative spread ∆( w − w p ) w − w p is independent of initial w in a multiplicative process. Also, while a multiplicativeprocess resembles an interest rate, we should not followthis analogy, as an interest rate is a drift, not a spread(see [26] and [27] for calibration issues with GBMs). Byconsidering the width of the distribution at a typical timeof economies, t = 1000 days, about 3 years (and less thantypical periodicity of economic cycles, say 8-40 years), wechose to home in on relatively large variations [3, 8]: Us-ing β = 0 .
06, we have ν drift = − . σ ( t = 1000 days = 1 . e σ = 2 .
92 shouldbe applied to ( w − w p ), whose most frequent or medianvalue is expected to be closer to w p than to w . So wedescribe, over a duration of 1000 days, a spread from,say w = 600 ( w = w p + 200) to the 1-standard-deviationinterval [400 + 200 e − σ ,
400 + 200 e σ ] (cid:39) [469 , N agents. We pinpoint the role of the wealth-iest agents in determining the overall fate of the en-semble [1, 3, 5, 15, 18, 30]. Since we have an analyti-cal form of the wealth distribution, we can deduce the FIG. 2. (Color online) Log-log plot of N/k-iles x N/k ( t )(dashed lines) as a function of time for N = 3 600 , β = 0 . x N/k ( t ) is ( k/N ). Thecurve k = 1 is superimposed as a solid dark magenta line.The largest of N wealth lies, around and above x N ( t ), plottedwith an added solid line. The largest wealth from several sim-ulations are shown at selected time points (green dots), thefraction of points above x N/k ( t ) at a given time t showing theexpected rarefaction trend with decreasing k . statistics of the wealthiest agents at time t [31]. We donot embark on this exercise rigorously [15], though, weonly remark in passing that extremes are key to non-ergodicity demonstrations (Eq. (7) in Ref. [23]), butrarely explicit. We can get enough indication of thelocation of the maxima by a simpler use of the Gaus-sian normal distribution [31]: if we have N realizationsof a Gaussian variable, we have a good approximationof the statistics of the largest by slicing the Gaussianinto N slices of even weight (see work on electron re-laxation bottleneck in quantum boxes [33, 34] for a sim-ilar use of a Poissonian statistics). The N -th slice isan acceptable approximation of the distribution of thelargest wealth, in spite of its abrupt cut-off, avoidingthe more tedious math of the theory of extrema [15, 31].What is interesting for us is the ability to use the com-plementary error function erfc( x ) and its inverse erfc − to deal with the main aspect of such statistics. This issimpler to grasp, for those less familiar with extremallaws, than using the exact Gumbel law, i.e. expressingthe cumulant U ( x ) of the largest value distribution in aform often denoted U ( x ) = exp( − exp( − z ( x, N ))), with z ( x, N ) = [ x − a ( N )] /b ( N ) with analytical expressionsof a ( N ) and b ( N ): the typical error by taking this naivemean instead of the exact one is ∼ . e . − ∼
20% error.With just a little more generality, we look for the edge x N/k of what we can term as the
N/k -th slice, with k = 1for the largest, k = 2 for the two largest, etc. in the spiritof quantiles [15, 18, 19]. And we also allow ourselves to use a fractional k , e.g. k = 0 .
25 in
N/k in order to targeta range reached by the largest variable only in 25% ofthe statistical events. We shall call these approximatequantile boundaries those of the
N/k -iles, generalizingon deciles or centiles, notably in the plot of Fig.2 below.Specifically, using standard Gaussian statistics, we iden-tify the moving edge x N/k of the
N/k -th slice accountingfor the
N/k -iles statistics such that (cid:90) ∞ x N/k ( t ) ˆ P ( x, t ) dx = kN (8) x N/k ( t ) = ( x + ν drift t ) + √ σ ( t ) erfc − (cid:18) kN (cid:19) (9)Note that the second formula takes into account thecenter drift. If we run a simulation of N agents duringa long enough time, we expect x N/k ( t ) to first feel theinfluence of the diffusion and the erfc − function. In alog-log scale, the largest element ( k = 1) is the upperenvelope of the set of the N x j ( t ) = log ( w j ( t )) traces.If we neglect drift, at short times, we have x N/k ( t ) − x ∝ √ t = exp(log( t/ k , namely the coefficients erfc − ( k/N ) whose trendagainst k is logarithmic. This can be seen on the left ofFig.2, for N = 3 600, where a set of several k values isrepresented, with k = 1, i.e. the N -ile of the Gaussian,shown as a magenta solid curve superimposed over theset of dashed-lines for other k values.Now, at long times, since we have a negative drift,as illustrated by Fig.1a, even though the standard devia-tion grows, the average locations of the N/k -iles must all,sooner or later, shift left to x → −∞ . More precisely, thesmaller k the later the trend of increasing x N/k ( t ) (i.e.diffusion) is reversed by drift to a decreasing one [15].This is the essence of the mechanism allowing the break-down of GBM ergodicity as noted in [23]. In other words,it demands too small a fraction of the sample (much lessthan N − , thus less than one agent) to get a chance ofrealizing high values in the tail, even though such valuescarry a major contribution to the (unbound) expectationvalue in a continuum view.We have added on Fig.2 the plot of extremal valuesat selected logarithmically spaced times drawn from aset of eight numerical simulations up to tmax = 55 000“days” (about 150 “years”), using the resident randomMATLAB generator. Let us comment for instance theoutsiders which are the highest points situated aroundthe “N/0.01-ile” curve. Since we chose to sample about75 points per curve, hence 600 points for 8 curves, thehundred-times rarefaction vs. k = 1 entails, probabilis-tically, a number of points around the N/ . t = 20 000 ±
15 000, where these outsider points are
FIG. 3. (Color online) Average wealth of eight simulations(solid lines of different colors) with the same parameters N =3 600 , β = 0 .
06. The initial Brownian-like motion around w (level indicated by the right-side dashed line) is graduallysuffering larger and larger fluctuation, essentially associatedto the large x N ( t ) at intermediate times ( t ∼
15 000 −
30 000).At larger times, the drift eventually dominates and the aver-age wealth is stuck to the poverty level w p (indicated by theleft-side dashed line). more clearly seen than at earlier times, we are concernedwith a subset of about 160 points, and we find around 2to 4 points in this subset instead of the 1.6 expectation,a reasonable amount for a random draw of this kind (fac-toring also our approximate extremal law).The above exercise is useful to grasp how the Gaus-sian statistics can be sampled along a long time series:we may, at some times, and provided that we are notrestrained by correlation (hence at the lower limit, notat the scale of two adjacent times with w ( t + 1) and w ( t )separated by less than a factor β ) [28], reach values muchhigher than x N ( t ) in a given simulation.Let us now focus on the global wealth. Although thestatistical average of a single operation is zero, actual op-erations have some nonzero average. This emphasizes therole of discretization [9, 15](again, ergodicity breakdownis the overarching issue [23, 25]). At the start, with all w ’s of the same order ∼ w , fluctuations of this origincancel out reasonably well: as N − / at a given time.Further along the time series, they pull also randomlyup or down. So for some time after the start, the averagewealth gently performs a Brownian random walk around w (and the total wealth around N w ), as seen on thevery left side of Fig.3 on a linear time scale.But, as time goes and the largest agents wealth samplesvalues around x N/k ( t ) for k = 1 easily, and further ateven smaller k values, large fluctuations are introducedon the total, and thus on the average wealth. This is the“tail of the dog”, but here the tail is not “wagging thedog” forever, as actually there is independence of the N agents wealth, so the stronger fluctuations of the averagewealth only reflect the inescapable maximum regions of the curves of Fig.2, when the diffusion to large wealthis compensated by the slow drift in the x -space (also anonset of apparent ergodicity breakdown). The β scalingof the drift velocity shows how discretization, and thus β , is a significant (but not critical) parameter.We can deduce the order of magnitude of most quan-tities as a function of N and β , the sole relevant param-eters at this stage, using the approximate drift and theapproximate extremal law: x N/k ( t ) (cid:39) (cid:18) x − β t (cid:19) + (cid:114) π β √ t erfc − (cid:18) kN (cid:19) (10) t max N/k (cid:39) πβ (cid:18) erfc − (cid:18) kN (cid:19)(cid:19) (11) x max N/k (cid:39) x + 3 π (cid:18) erfc − (cid:18) kN (cid:19)(cid:19) (12)where the two last lines point the maximum of the firstline. For instance in our case N = 3 600, we obtain forthe N -ile position: t max N (cid:39) x max N − x (cid:39) .
31 =2 .
74 log(10), meaning that the edge of the richest N -ileslice is around ( w − w p ) ∼ . ( w − w p ) = 550( w − w p ) (cid:39) .
30 10 (the crude approximation erfc − ( u ) (cid:39) [ − log( u )] / is thus too coarse).Hence, we can also understand that the typical timeframe of the large intermittency window can be as-signed a practical interval such as [ t max N , log(100) t max N ] (cid:39) [10 520 ,
48 530], with an upper boundary taken here so asto be exceeded only in one out of 100 cases: the statisti-cal character of this upper boundary is apparent throughthe presence of a smaller but clear isolated intermittencypeak at t ∼
42 000 ∼ log(50) t max N in one of our eight sim-ulations. The typical time scale of the intermittenciesis more difficult to provide [8, 10, 11], but it is logicalthat it appears as only a fraction of t max N because it takesless time than this for extreme fluctuations to enter andleave the extreme domain ( t max N is the time to go fromaverage wealth to extreme wealth for the fastest of N elements). From the point of view of realism [3], since t max N is about 30 years, we have here a confirmation thatour β = 0 .
06 daily value is not that large: this time scaleof 30 years is not incongruous with that of actual majorcrisis in capitalist economies.Last, the typical large deviations of the average areon the order of a few times the naive quantity ( w − w p ) exp( x max N ) /N found when counting the role of asingle wealthy agent as causing the fluctuation in themean: this quantity associated to k = 1 is small,( w − w p ) × / .
7. But we have enoughtime in one run, given the relatively flat situation around t max N , to sample rarefied maxima with smaller k , typically k ∼ .
25 in one run. Then the above quantity becomes( w − w p ) exp( x max N/ ) /N , which is about 2 000. On few(statistically on one) of our eight runs, we can of courseexperience eight times scarcer cases ( k = ) reaching amaximum average at ∼
12 000.We now comment the β = 0 .
06 connection with N . Ifone takes for N the population of a large city, N ∼ ,then, since 3 600 ∼ . , we get at given β essen-tially a doubling of x max N and t max N since (cid:2) erfc − (cid:0) N (cid:1)(cid:3) ∼ log( N ). So if we wish to retain the same character-istic time, a couple of decades, we have to modulate β = 0 .
06 by a factor √ N ∼ would demand a factor of 2 [3, 6, 8]).Having explored an ensemble of N perfectly non in-teracting agents, we next implement a mechanism forthe tail to be “wagging the dog”. Overall, the dynam-ics of the global wealth of our simple non-interacting setof N GBM epitomizes how delicate it is to describe theregion of largest manifestation of ergodicity breakdown.We conjecture that clarifying its dynamics would be help-ful for the understanding of related models.
III. THE N-AGENT MULTIPLICATIVE MODELWITH RESET AVERAGE
We now consider a feedback mechanism that consistsin resetting the average at its initial w value at all times.This will therefore introduce correlations [28] and will im-print the intermittent dynamics of the total wealth ontothe whole distribution, paralleling the way emerging so-cial structures affect all corners of society. The abovemodel of independent agents with its indefinite downwarddrift that accumulates all agents wealth to the poverty“floor”cannot inspire even a stylized description of eco-nomic reality. We do not want to affect the multiplicativeaspect (the GBM), however. Technically, we simply im-pose, for all agents j : w j ( t + 1) = λ ( j, t ) w j ( t ) × (cid:80) Nm =1 λ ( m, t ) w m ( t ) N w (13)where we made explicit the random variable draw forthe j-th agent at time t, following the distribution law ofEq.3.Most usual discussions of “normalization” in agent-based simulations are about the growth rate. They arealso invoked in several works on random Brownian mo-tion and thus GBM, but we could not find the conse-quences that we find here [1, 3, 5, 9, 13, 15]. We shall dis-cuss in Sec.V the socio-economic meaning of this choice.In our model, it is most instructive to visualize the fateof the extreme wealth and that of the whole distributionunder this new assumption, as we propose through Fig.4and Fig.5(a-c) respectively. Of course the word “fate”means here all the coupled dynamics of society and in-equality, with its distribution of events, correlations, andcharacteristic times.In Fig.4, we see that under the new assumption ofmean normalization, once large wealth are obtained, thedownward drift is canceled: the maximal wealth remain FIG. 4. (Color online): Same log-log plot of maximum wealthvs. time as Fig.2, in the case of reset average, i.e. forcednormalization of the total wealth to
N w . After reachingthe point of maximum wealth expectation without reset, thepermanent regime retains a fluctuation pattern similar to thatoccurring around the maximum. around the established level x max N , and display large fluc-tuations. We thus operate permanently at the brink ofergodicity. We give a microscopic look into the distri-bution of wealth thanks to Fig.5(a), a color map of his-tograms fabricated at linearly spaced times t (spacing∆ t ∼
30 days), by integrating over ∆ t .We can now see that there are collective collapses of thewealth distribution core, down to values close to w p , assoon as there is a chance that the largest value reaches x max N , say from t ∼ x max N can be reached for the first time at a moderatefraction of t max N , not surprisingly from the general aboveanalysis of Sec.II).Also striking is the fact that these collapses are fol-lowed by revivals, some of them as developed as the startsequence (where we remind that all w j start from w ),with an overall intermittency pattern. Since we have seenabove that there would be occurrences of large averagewealth deviations, and that they are due to very few in-dividuals down to a single one, we infer that the samemechanism works here: once an agent is becoming thewealthiest in a steady enough way (that perturbs onlymarginally the distribution), it suffices that this wealthyagent undergoes larger fluctuations to induce an over-all fluctuation of the masses (see the treatment of firmsdeath in Ref. [15] with extremal law statistics: it providesa resembling, but not directly comparable pattern). Thatis what we picture as “the tail wagging the dog”.In terms of current analysis of GBM-based economicsmodels, we should be looking at time-constant distribu-tions of different classes of agents, and at their corre-lation. This would be a highly rewarding analysis if theresulting dynamics can be correlated to the available eco-nomic data through not only the trends of inequality, but FIG. 5. (Color online): Illustration of a typical sample ofwealth w N ( t ) under the reset-to-average assumption. (a) His-togram of wealth using a few hundred log-spaced bins, show-ing intermittency at multiple time scales; (b) the sorted tra-jectories of w j − w p for selected w j ’s. Note the correlationamong the majority of small wealth, but note that there israther an anticorrelation among this vast majority and thetop three wealths; (c) Gini coefficient of the distribution vs.time, with large random fluctuations that still bear the sig-natures of intermittency of the few wealthiest agents. also economic, sectoral and territorial discrepancies. Thefact that we can see a “tail” and a “dog” also points thepossibility to distinguish social classes and their line ofdivide, as will be developed briefly.The detail of the wealth fate can be perceived inFig.5(b), where we plot on a log scale a selection of thesorted temporal profiles of w j ( t ) − w p . We clearly seethat the troughs apparent in the tail of the poor agentscorrespond to the aftermath of a peak of the very fewwealthiest agents. In other words, in a zero-sum ex-change game due to the fixed mean wealth, any of thelarger-than-average fluctuations of the wealthiest is feltby essentially all agents, and can be felt as a big shock.Let us present our distributions of wealth as a pre-ferred economic indicator. Although fundamental ones have recently been proposed in relations with GBMs[24, 25], we present in Fig.5(c) the well-known Gini coef-ficient [29, 35]. Its variations are clearly triggered by thefew wealthiest agents. The curve shapes are not identical,but the major peaks, troughs and shoulders are clearlycorrelated. These curves contain both (i) the dynamics ofinequality in terms of distribution of time constants, in-sofar as a picture of subsets with a reasonable stationarydistribution of time constants applies, (ii) an indicationof the amount of correlation within agents, as they ac-count for the amplitude of the fluctuation.As for the possibility to define subsets, since we haveseen that the largest fluctuations are the leading events,we try below to define two dynamical “classes”, separatedby a “water divide” line of wealth flow. Such a picturemay provide an account of the actual GBMs mechanismsand invite resonances for the stylization step betweenmodel and economic reality.Specifically, we elaborate a color map of a matrix de-scribing the flux pattern between agent pairs ( i, j ), as isdone in Fig.6. Mathematically, we work on the sorted se-ries of wealth: we take first the product of the time seriesof the sorted wealth derivatives, y j = dw j /dt , and wethen fabricate a log-type indicator of the absolute value,but we keep track of the sign of the product to distinguishbetween gain and losses: A i,j = t = t max (cid:88) t =1 (cid:104) w ( s ) i ( t + 1) − w ( s ) i ( t ) (cid:105) (cid:104) w ( s ) j ( t + 1) − w ( s ) j ( t ) (cid:105) ≡ (cid:90) t = t max t =0 (cid:34) dw ( s ) i dt (cid:35) (cid:34) dw ( s ) j dt (cid:35) dtC i,j = sign( A i,j ) [log( A i,j )] , (14)where the superscript ( s ) denotes the sorted ensemble(dynamical sorting at all time t , so it scrambles the actualagents throughout the simulation time).In Fig.6, we clearly see a negative correlation betweenthe first four agents ( i = 1 to 4) and all agents beyond j ∼ i, j )-corner at the top-left of this matrix, shows that till values of the indices( i, j ) ∼
12, the correlation is still not clear cut, althoughthe trend toward positive correlation increases for smaller( i, j ). This is a signature that we have a fairly abruptpartition between the wealthiest and the mass (a massthat includes the 10 −
100 “affluent” wealthiest out of N = 3 600), the former influencing the overall fate byattracting wealth from all other agents [3, 5, 11, 12, 18].Thus, our indicator tells how to define two “effectiveclasses”, separated by the “water divide” of wealth flow,on the average. The fact that it is not a single agentis of good omen for the application of the model. Play-ing with the basic GBM parameters ( N, β and the ratio w p /w ), the relative size of this class would evolve fromthis ∼ .
1% to another fraction. The inner fluxes inside
FIG. 6. (Color online): Correlation analysis on ranked wealth.See text and Eq.14 on the particular correlation-based indi-cator C i,j used here. We represent as a color a quantity mea-suring the fluxes and their signs, between the affluent and thepoorest of the agents, with a step-wise logarithmic samplingof the 3 600 × ∼
40 wealthiest agents. each class could be studied in order to further partitioneach of these “basin” of the water divide picture. Then, areverse procedure could help establishing sensible cleverGBM nonlinearities, for example a β ( N, w ) dependence,that could help mapping actual econometric data setsinto GBM models with somehow socially agnostic as-sumptions. The dynamics of each subset equally deservesattention [25–28].What we can do with modest effort is to examineFig.5(b) in more detail to get qualitative clues on thedynamics. If we look at the lower part of the distribu-tion, well below the “water divide”, the large intermit-tency features are more and more quickly washed out aswe go to small wealth, and the main governing factorseems to be the negative impact of the aggregate wealth.If we now look at the population across all times, inFig.6 color map, we see that, as is logical, the indica-tor tends to vanish for the least wealthy agents (blue-green shades at the bottom right). The absence of ran-domness of the indicators sign occuring for the small-est wealth suggests that their inter-exchange (that takesplace in principle through the forced averaging processwhich includes their own collective fluctuation) is a mi-nority mechanism. Their fate is dominated, as is obviousfrom the overall distribution in time, by the influencesof wealthiest: the trickle-down effect in recovery phases(when the wealthiest give or “emit” wealth) or the aus-terity effect in collapse phases.We have now examined a canonical version of our mul-tiplicative wealth model. Once sufficiently large wealth are generated, there is a regime of boom and busts, withabrupt collapses and slower revivals, due to the couplinginduced by the constant average. Fluctuations of therichest are enough to cause large swaths of the popu-lation to be affected within short times. More analysiswould entail tools such as momenta or Laplace trans-forms, with a scope of finding “excited modes” of thedistribution. While this has a simple sense around equi-librium, we have no clues as to what are excited modesin a deeply nonstationary and broken ergodicity context.However, if we tame the nonstationarity, we may recovera system amenable to an eigenmode (fundamental andexcited modes) analysis. The idea would then be to trackhow these excited modes behave when reintroducing non-stationarity. Taming nonstationarity and inequality in-termittency is just the purpose of the following.It is tempting to think of “nudging” the underlyinglaws so that this intermittency regime and its inducedcollapses are avoided. This entails avoiding the advent oflarge wealth if we want to maintain that β itself repre-sents a “psychological constant”, the amount associatedto risk at daily scale (scaling with the square root oftime, basically). Forbidding large bets to owners of largewealth per se, even when they are designed well withinregulatory barriers, would be a too directive way to in-terfere with the economic microscopic decisions. So inSec.IV below, we introduce a modification of the basicprobability law of our multiplicative process, Π( λ ), thataverts the build-up of “extremely extreme” wealth. Sta-tionarity is viewed as the obtainment of a fundamentalmode of the system, with the corollary that obtainmentof excited modes in the same frame is natural, but weshall not study their dynamics in the present work. IV: AGENT ENSEMBLE WITH RESETAVERAGE AND WEAKLYWEALTH-DEPENDENT MULTIPLICATIVEPROCESS
To avoid the advent of large wealth, we first define a“status indicator”, based on wealth here. The role of“status” as a general factor in setting the price of ex-changes dates back to Aristotle and was revived in socialscience and anthropology by P. Jorion [20, 30, 32], withthe aim of escaping the conventional wisdom of pricesfluctuating around a “fundamental price” that an undis-torted market is supposed to reveal (see also the finaldiscussion). Aside such general views, a “status indica-tor” would be a good channel to link in the future ourdeliberately limited study to more complex ones with adeveloped social account [4, 18, 22, 29]. We cannot usethe “intrinsic” classes defined by the “water divide”, be-cause they correspond to time average of a nonstationaryprocess, so that they are not known until the relevantfluctuations did take place. We thus find it sensible to0define a continuous status without connection to the non-stationary dynamics. Here, our status indicator denoted S j is a simple homographic function based on the com-parison of wealth above poverty to average wealth: S j ( t ) = w j ( t ) − w p w + ( w j ( t ) − w p ) (15)So it tends to unity for large wealth, is one-half for w = w + w p ( w = 1 400 in our case), and tends to zero for w → w p . We then modify Π( λ ) to introduce a counter-acting bias. Specifically, we make Π( λ ) status-dependent( S -dependent) using a skew factor ε as follows:Π ε ( λ ) = Π ( ε, S ) rect (cid:18) λ − (1 + εS )2 β (1 + εS ) (cid:19) = 12 β (1 + εS ) rect (cid:18) λ − (1 + εS )2 β (1 + εS ) (cid:19) (16)So Π ε ( λ ) is centered at its centroid ¯ λ = 1 + εS , span-ning uniformly the range 1 − β → β (1 + 2 εS ). Techni-cally, we obtain it as (1 + β [1 − εS ) instead of1+ β [1 − , ε plays the role of a wealth amplifier if ε >
0: the wealthiest entities turn exchanges to theiradvantage, a well-known fact, evidenced by Piketty usingthe yields of the funds of US universities [19]. It playsthe role of a taxation mechanism if ε <
0, pushing theaverage factor ¯ λ to more than unity for the poorest andless than unity for the wealthiest.We shall see that, surprisingly, very small skew fac-tors ε are sufficient to avoid the build up of large wealth.Or not so surprisingly, as it boils down to inhibiting thegrowth of the very few large wealth agents of the his-togram that was built up in years, not days. If thisdistribution is diminished at large wealth by a modestfactor per octave, then, since concerned agents are 10 ormore octaves wealthier than the median (2 ∼ . factor that gave the maximumN-ile at 550( w − w p ) (cid:39) .
30 10 ), it provides a signifi-cant inhibition of the tail. We will report below effortsto quantify the stationary distribution that results fromthis modification. Let us examine the impact of ε (cid:54) = 0through simulations first, using all other parameters asbefore.In Fig.7, we examine the wealthiest agent evolution forvarious skew parameters ε . We have left as a guide thecurves of the ( N/k )-iles of Sec.II. We see that the modifi-cation does have the expected effect, and that this effectis large even for ε values as small as ε = − . ε = 0 ina situation of extreme wealth reaching a large fractionof the total wealth: there is little room to expand morethe wealth, and the distribution clearly saturates (akin FIG. 7. (Color online): Fate of the richest wealth under dif-ferent assumptions for the skew parameter ε . The analyticalcurves provided for the simple case of no skew and no resetin Sec. II are left as a guide to the eye.FIG. 8. (Color online): Gini coefficient under different as-sumptions for the skew parameter ε as indicated. to wealth condensation) for the ε = 0 .
03 skew (positivefeedback) parameter shown here. For negative skew pa-rameters ε , we see that the wealthiest values clearly di-minish by about 1.5 decade for ε = − .
03. At the sametime, relative fluctuations tend to diminish (graphicallyobvious in log scale).In Fig.8, we show the Gini coefficient evolution for var-ious values of the skew parameter ε . The largest fluctua-tions are those of ε = 0 .
03, but, as said, they “bump” onthe ceiling of wealth saturation. Also, due to the posi-tive feedback and the subsequent roll-off of large wealth,variations are very quick. For negative values of ε , now,fluctuations of the Gini coefficient diminish clearly evenfor ε = − . ε = − . ε = − . FIG. 9. (Color online)(a-e) For a simulation with a small negative skew parameter ε = − . .
03, with extreme intermittency; (e) shows the case ofa large negative skew parameter − .
03, with a very steady situation. an interesting limit in our simulations window (a win-dow intended to describe boom-and-bust cycles at the ∼ t = 36 000and t = 45 000. At these points, the wealthiest agentclearly exhibits an anti-correlation with most others, andshakes the whole distribution, see Fig.9(b), as identifiedon similar earlier graphs. These are the points of surgingGini coefficient, as is seen on Fig.9(c), albeit by a mod-erate amount. For the sake of comparison, we provide onFig.9(d,e) histograms of the two extreme and contrastedsituations ε = ± .
03. In the positive case, we see thatthe evolution is a tale of few moments of “shared pros-perity” and many moments of utter inequality. But assoon as an agent takes over, it acts over the whole distri-bution, and it is “wagging” all the distribution very soon( ∼ C i,j defined above as in Fig.6, but for a negativeintermediate value of the skew parameter ε = − . w , but we have not ex-amined this in detail. There are a few spurious positivecorrelations on lines next to the diagonal. We believethey stem from the choice of using an indicator based onsorted distribution. The sorting introduces correlationbetween adjacent ranks, if they both mostly suffer fromother agents, but just spend some time “crisscrossing”.Anyway, it does not perturb much the overall picture,but rather indicates that the next steps in such simula-tions would be to understand, as in many current physicsproblems, the role of correlations [11, 12, 22, 29], whichis a general concern in the newly emerged considerationsof GBMs [25–28].In Fig.11(a) we show the histogram of wealth distribu-tion at t = 55 000 for various simulations. Depending onthe skew parameter ε , we see clear indices of the mech-anisms operating for these different distributions. For askew parameter ε ≥
0, we see strongly populated peaksthat “scar” the left side of the distribution, some of themnot so far from the main peak. The distribution is broad,and we also see on the large wealth tail a few peaks withone or a few individuals that are above the trend of thetail at lower values. Both indications are logical with the2
FIG. 10. (Color online) Map of correlation of wealth varia-tions, C i,j , among sorted agent records, for a negative skewcoefficient 0 .
015 as in Fig.6. The transition from positive tonegative fluxes is smooth and lies around i, j ∼ mechanism of “the tail wagging the dog”. We see nowsomewhat more in detail that there are highly populatedsets of agents that were in some narrow interval close to w p , and that benefited from an upward kick when thewealthiest agents fluctuated downward. The statisticalcharacteristics of these intermittent bunches may be aninteresting part of future work, in relation with GBMdynamics.As for the distributions for ε <
0, they clearly getnarrower as ε becomes more negative, and logically, theytend to become stationary. The equilibration time [8, 26,27] is shorter for the more negative values of ε . We nowhave a clearer view of how the distribution is curtailedon the high end: by a decade or so around w = 10 , forthe case ε = − . vs. the reference ε =0. The factthat such a small skew could avert the large fluctuationsinitially surprised us (somehow as the diverging feedbackof inequality in the reallocation+GBM model of Ref. [27]surprised their authors). But considering the effects ofthe residual drift that we explored in Sec.II, it is not sosurprising that a very limited but “daily” drift acts insuch a large manner.Since a stationary distribution results, we can deter-mine it, thus allowing comparison with the broad litera-ture addressing this topic [6, 7, 9, 11, 14, 26, 28, 29, 36].It is possible because we can assume that the normaliza-tion mechanism does not play a role anymore [13, 14].The signature of this mechanism was the set of peaksor bunches on the left side of the distributions. Theyare still seen in the limit case ε = − .
005 and are as-sociated to very modest wealth steps values (less thanunity, hence ∼ ( w − w p ) / P ( w ) = (cid:90) + ∞ w p P ( w (cid:48) )Π ε ( λ ) dλ, (17)which accounts for a single time step in a mean-field view,assuming as a boundary condition that no probabilityflux comes from the region close to w p . We should alsocare that the probability Π ε ( λ ) is actually coupled to w by the status factor. A full notation would be Π( w (cid:48) , λ ) orΠ( S ( w (cid:48) ) , λ ). However the numerical values in this equa-tion are spread on decades. So we can transpose this inthe { x, (cid:96) } space, whose (now stationary) probability lawand multiplier law are respectively ˆ P ( x ) and π ( (cid:96) ). Withthe now additive algebra, given that x (cid:48) + (cid:96) = x , and withan adequate redefinition of π ( (cid:96) ) into π ε ( (cid:96) ) to incorporatethe status S , we find:ˆ P ( x ) = (cid:90) + ∞−∞ ˆ P ( x − (cid:96) ) π ε ( (cid:96) ) d(cid:96) (18)This form is reminiscent of a convolution (as it should).But due to the dependence of π ε ( (cid:96) ) on x through thestatus S , it is not a convolution. Nevertheless, it is pos-sible to solve for this equation in the form of an eigenvalueproblem onto a discretized and uniform set { x m = m ∆ x } of M values of x . Also, in the limit of a large enoughnumber of iterations, we can remember that the fateof the distribution was given at ε = 0 by its drift andsecond momentum (diffusion constant, essentially β forus) [1, 2, 9, 15]. While we do not have a theorem toextend this to the case ε (cid:54) = 0, we conjecture that thefirst-order findings we want can be made retaining thisassumption. So the matrix representing Eq.17 as an op-erator on ˆ P is built up as a Toeplitz matrix, with the n -th diagonal having a coefficient π ε ( n δx ) . This intro-duces a constraint as we want to span several decades(12 decades, see below) , so that even in a large ensemble { x m = m ∆ x } with an M value of a few thousand, onlya few of these values fall within the modest range locatedbetween extrema of (cid:96) , essentially log(1 ± β ), apart fromthe small corrective action that describes the law π ε ( (cid:96) ).However, with some care on this sampling, and giventhe assumption mentioned above, we could find signifi-cant solutions with matrix sizes M of a few thousand,and typically 10 −
20 filled diagonals. Then, a significant π ε ( (cid:96) ) can still be put up by operating on the rows of thematrix, shifting the centroid of the π ε ( (cid:96) ) distribution tothe proper drift-induced value equivalent to Eq.16, butnot taking into account the modified width of the distri-bution. This introduces some second-order effects whenit comes to converge to a stationary distribution.3 FIG. 11. (Color online)(a) Histograms of ( w − w p ) on log-log scale, for various skew parameters as indicated at a largeenough time to escape the initial phase. Note the spikes on theleft side of the distributions down to parameter − . Fig.11(b) shows the result of this approach using a setspanning 12 decades. We indeed find that the largesteigenvalue of our matrix is nearly unity, and its eigen-vector is generally a bell-shaped distribution. For tooweak negative values of ε , about | ε | < . x values. Just becausewe do not normalize the rows of the matrix at its “cor-ners”, the probability can be “dissipated” there, and thedrift+diffusion processes must accommodate this numer-ical boundary. Such a typical appearance, provided herefor ε = − . ε , we find stationary bell-shaped curves. They present most of the characters of theactual distribution: there is notably a shift to the rightand to a narrower distribution that is quite comparableto the simulation of Fig.11(a). However, we suspect thatour solving procedure is not accurate enough to attempta meaningful fit: a better numerical solution should besought. Even without an exact account of our model, wecan nevertheless discuss its benefits.An interesting exercise around the class of station-ary distributions that are currently under scrutiny is toattempt to look at the relaxation rates of the excitedstates [27]. Very plausibly, the relaxation rates will befaster for the higher excited states, and the first excitedstate [26] gives a measure of the most relevant time scalefor external shocks that affect inequality, i.e., that affectdifferentially the rich and poor agents. Nonlinearitiescould also be investigated in terms of the effect of cor-relation. Naively, letting an initial state evolve from thetwo linear combinations | p (cid:105) ± | q (cid:105) of the p -th and q -thmodes could modify the relaxation rate due to the sta-tus term, as this latter is not respecting any of the modeorthogonality conditions. V. DISCUSSION
The quest for econophysics models to understand in-equality recently evolved from the study of stationarydistributions [2, 4, 5, 12, 14, 16, 18, 19, 22, 29, 30] to themuch more fascinating issue of the distribution dynam-ics [25–28]. The simple tool of GBM gives insight to suchmodels, not least because it addresses some limitation ofmainstream economics (e.g. a bounded utility function),but also because it helps tackling the false intuitions thatarise when ergodicity breaking is not properly taken intoaccount [23, 24].Setting up a model with a finite number of agents anda fine-grain (“daily”) time discretization, we have intro-duced a nonstationary regime of sustained intermittencyby using the normalization of the total wealth. A livelydynamics emerges, with still much to analyze. The num-ber of agent used in the simulations ( N = 3 600) wasenough to attain in a reasonable time scale the situationof an extreme degree in wealth capture by a few individ-uals, whose decisions then impact the fate of all agentswithin short times (say, few months), but not in any cir-cumstances, rather only once a crisis is triggered [15].The way the fluxes have their signs undergoing inversionwhen scanning in a sorted agent distribution (Fig.6 andFig.10) is one of the most meaningful signatures we got.It defines two natural subsets across a “water divide” ofwealth flow. Their coupled fate can then be capturedin a nutshell by the “tail wagging the dog” metaphor.It could be applied to actual statistics or to any of the4many more explicit models. Sticking to the dynamics ofsubsets and aggregates, we would have tools that remainintrinsic or “agnostic” enough along this line. We arealso aware that calibration of GBMs is in infancy andwill by itself reveal several features of interest or evenprompt new uses of GBMs.Also, generally speaking, when inside a general largesystem, a subsystem presents sufficient stationarity, itsdegree of redistribution could be studied with an appro-priate scaling of N [6, 8, 9, 12, 15, 16](we considered onlyGBMs coupled by the normalization, but an absence ofreallocation in the sense of [27]). The comparison isnot limited to wealth as traditionally quantized in eco-nomics, it can be adapted to various cases around thegeneral balance idea. This principle entails the statis-tical fairness in the microscopic trend (as much chanceto get more than less in a single event), but neverthe-less the small gain of a minority in relative terms ap-pears to be self-amplifying, even weakly. This could befor instance the fate of fashionable topics in a domainof science, where the developing trend first looks like afair reward, but if it becomes a dominant trend, it canlead to too many followers and production of apparentknowledge with little actual relevance in a majority ofcases [35]. As several of these domains are not as long-lived as human economies, it makes sense to start simu-lation from an apparently stationary distribution. Then,the issue of dynamics has a simple focal point: how muchthe first main crisis can be anticipated (see the rich stud-ies on firm births and deaths [15]), or more precisely: canwe find how the onset of crisis can be described in a moredetailed fashion, e.g. by connecting its probability to allaverage features/momenta of the wealth distribution orof its underlying log-scale counterpart ?In the economic domain, the large intermittency is of-ten linked to the Black Swan paradigm of Nassim. N.Taleb, which relates rather to bust phases, but it is moredifficult to assign “white” or“withish” swans in the pos-itive boom phase of cycles among the impact of techno-logical, societal or political changes. Microscopic studiesof different sectors and their interactions could benefitfrom the a comparison with our kind of stochastic modelin this respect.After such “microscopic” considerations, let us takebriefly a broader perspective: Our initial intention, thatwe hope to be still present in the result, was to putPiketty’s historic-economic narrative [18, 19] and someof its “obvious” consequences into a model that wouldgo one step beyond the stage of “riches become richer”,with the further prospect of shedding light on how the un-derlying networks and their concentration effects operatein terms of statistical distribution of economic and socialvariables [22, 30, 32]. On this way, we were faced with thefact that growing inequalities and nonstationary distri-butions occur even in the simple paradigm of apparentlylocal fair exchange, a result that stems from the natural drift of a zero-sum-exchange in log-scale terms. Therewas not much appearance of such issues in the econo-physics literature until a few years ago. Then, as theissue of evolving inequalities became more paradigmaticwith data available across the 2008 crisis, the dynamics ofinequality and the capability of GBMs to describe thembecame a center of intense attention [23–28]. Indeed,there is a interesting parallel between on the one handPiketty’s “divergence” of the r > g picture, whose mean-ing is rather historical than a precise econometric exercise(hence Piketty stops short of a divergence model), and onthe other hand the nonstationarity and ergodicity break-ing of GBM ensembles [23–25].Taming this nonstationarity involves no less than as-sessing whether our economies run in a near-equilibriumfashion, or more deeply out-of-equilibrium [27] eventhough sociology and economics can track a number ofslowly drifting items that are deceivingly suggestive of anadiabatic evolution restoring an equilibrium induced bythe noise of external shocks.In our case, instead of introducing an explicit taxa-tion mechanism having an extra variable and entailingno less than the prerogatives of a “State” to run it, wechose a more implicit or self-contained approach: Ourabove presentation of the equilibrating mechanism as re-lying on “status” is drawn from anthropological consid-erations brought into the realm of finance and economicsby Paul Jorion, who found that the law of supply anddemand was only marginally verified in actual commu-nities submitted to extreme risks of subsistence (fishercommunities for instance). Rather, based on Aristotleinspiration (picked up from Karl Polanyi’s writings), thesurvival of the community, and the reproduction of themember’s status throughout its social and economic ex-changes, was felt to be a more general factor, preventingprices to fall too low, or on the contrary, causing the buy-ers to have the last word even in periods of high demandas part of their accepted higher status.We have therefore introduced an equalizing mechanismakin to taxation directly as an exchange bias that can beseen also as an average price bias. The fact that withenough corrective strength, the distribution turns fromnonstationary to stationary is no surprise. The interest-ing point that would not have been guessed easily at firstis that a quite limited skew or bias, on the order of 1% inthe daily transaction, is sufficient to strongly suppress theadvent of inequality-induced crisis. It is admittedly notobvious to connect (and possibly contrast) a small dailybias on the one hand and, on the other hand, the currentconventional wisdom that yearly tax rates for the afflu-ent must lie somewhere in a range of 15-60%. At a timewhen economic models are under criticisms from severalpoints of views [18, 22, 29], we believe that the knowledgebrought by our simple model is a source of inspiration forall three communities of physics, econonophysics, and thebroader social sciences that embed economics. This in-5spiration rests on a fertile ground thanks to the recentconsideration of all GBM properties and of their subtleconsequences in inequality models [23–25, 28]. ACKNOWLEDGMENTS
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