Nonlinear redistribution of wealth from a Fokker-Planck description
NNonlinear redistribution of wealth from a Fokker-Planck description
Hugo Lima , Allan R. Vieira , Celia Anteneodo , Department of Physics, PUC-Rio, Rua Marquˆes de S˜ao Vicente, 225, 22451-900, Rio de Janeiro, Brazil National Institute of Science and Technology for Complex Systems (INCT-SC), Rio de Janeiro, Brazil
We investigate the effect of nonlinear redistributive drifts on the dynamics of wealth describedby a Fokker-Planck equation for the probability density function P ( w, t ) of wealth w at time t .We consider (i) a piecewise linear tax, exempting those with wealth below a threshold w , andtaxing the excess wealth with given rate, otherwise, and (ii) a power-law tax with exponent α > α > G .The introduction of an exemption threshold not always diminishes inequality, depending on theimplementation details. Moreover, nonlinearity brings new stylized facts in comparison to the linercase, e.g., negative skewness, bimodal P ( w, t ) indicating stratification, or a flat shape meaningequality populated wealth layers. I. INTRODUCTION
The tools of statistical mechanics to tackle out-of-equilibrium processes can be useful to describe the dy-namics of wealth [1]. The so-called random asset ex-change models [2, 3], assuming wealth transfers throughpairwise interactions between agents, have successfullyshown the endogenous emergence of stylized facts of realwealth distributions such as the concentration of wealthin heavy tails and the formation of a condensed layer ofthe poorest people [4, 5]. In fact, rich-get-richer mech-anisms can lead to the continuous accumulation of as-sets, accentuating wealth inequality, as observed in mostcountries over time [6–8]. Therefore, the effects of reg-ulatory processes capable of counteracting or mitigatingthese trends are worth of investigation. With this aim,agent based [9, 10] or probabilistic approaches [11] havebeen considered. Alternatively, the Fokker-Planck ap-proach is the natural counterpart of the stochastic dy-namics describing asset random exchange trading. BruceBoghosian [12, 13] derived an equation for the time evo-lution of the probability density of wealth P ( w ) thatemerges from the so-called yard-sale random transfers ofwealth, in which a fraction of a donor wealth is trans-ferred to another agent [14]. In this model individualsin an artificial society possess a certain wealth and par-ticipate in transactions by pairs, such that the total networth W , as well as the size of the population N , areconserved quantities, setting up a closed system. Theimpact of linear redistributive actions in such a closedeconomy has been addressed before [12, 13, 15–17]. Now,we consider nonlinear redistributive taxes, which can beregressive, progressive, or exempting the poorest popula-tion, and investigate how they can contribute to reducethe level of inequality.The remaining part of the paper is organized as fol-lows. In Sec. II, we describe the Fokker-Planck modelwith nonlinear drift and different settings that appearconcomitantly. In Sec. III, we present numerical resultsfor the stationary states and also for the evolution of wealth distribution, comparing the effect of the differenttax protocols, through the Gini index. Final remarks arepresented in Sec. IV. II. FOKKER-PLANCK DESCRIPTION
The stochastic dynamics of yard-sale exchanges withtaxes can be described, under certain approxima-tions [12], through an integro-differential Fokker-Planckequation (FPE) for the probability density function(PDF) of wealth, P ( w, t ), namely ∂P∂t = 12 ∂ ∂w (cid:18)(cid:104) w A + B (cid:105) P (cid:19) − ∂∂w (cid:0) f P (cid:1) , (1)with A ≡ A ( w, t ) = (cid:90) ∞ w P ( x, t ) dx , (2) B ≡ B ( w, t ) = (cid:90) w x P ( x, t ) dx . (3)The first term in the right-hand side of Eq. (1) can beinterpreted as a state-dependent diffusive spreading ofwealth, which has its origin in the microscopic randomexchanges between agents. In the second term, the drift f ( w ) represents the net gain (loss) of agents with wealth w . The derivation of this FPE with linear f ( w ) can befound in Ref. [12]. Now we assume that f ( w ) is given by f ( w ) = χ (cid:0) β ¯ w − γg ( w ) (cid:1) , (4)where γg ( w ) is the wealth raised from agents possessing w , through a nonlinear function g ( w ) times γ , a coef-ficient that can be time dependent, whereas β ¯ w is thewealth received back (the same for all people), propor-tional to the average per capita¯ w = (cid:90) ∞ xP ( x, t ) dx , (5) a r X i v : . [ phy s i c s . s o c - ph ] J u l via a coefficient β that can also be time dependent, andfinally χ controls the relative strength of the drift perunit time with respect to the diffusive process.If χ = 0, the PDF P ( w, t ) continuously evolves accen-tuating the condensation at w = 0 and the heavy Paretopower-law tail when w gets large [12], respectively mean-ing increase of the population at extreme poverty andconcentration of wealth among a small number of peo-ple (oligarchy). This endless process that accentuatesinequality can be broken by the flow of wealth from therichest to the poorest people, produced by the drift thatis present when χ > β = γ = 1 and g ( w ) = w [12, 15], meaning time-independent taxes pro-portional to individual wealth. We generalize the origi-nally linear drift in Eq. (4), by considering different ker-nels g ( w ) that are nonlinear (non proportional to w ).One of them is the piecewise-linear function g ( w ) = ( w − w ) H ( w − w ) , (6)where w is a positive constant, H is the step Heavisidefunction. The original version is recovered by setting w = 0 and β = γ = 1, in which case fees are proportionalto the individual wealth no matter how small it is. Wealso consider the power-law taxation g ( w ) = w α , (7)with α >
0. The taxation can be super-linear, i.e., pro-gressive ( α > < α < β = γ = α = 1.Besides the specific shape of the kernel g ( x ), there isanother issue that must be addressed concomitantly withthe drift nonlinearity. In all the analyzed cases, Eq. (1)conserves the norm of P ( w, t ) along time. It means con-servation of the population size N , whose distributionwithin the full range of wealth is given by N P ( w, t ).However, while in the linear case Eq. (1) naturally con-serves the average wealth ¯ w given by Eq. (5), otherwisewealth conservation must be imposed through the evolu-tion of the drift coefficients in Eq. (4). This is importantif we want to maintain the condition of a closed economy,where the total wealth W = N ¯ w is conserved over time,without losses or accumulation, for instance, by the regu-latory agency. We can let γ evolve in time, which meansan adjustment of the protocol (setting I), or vary β (set-ting II), meaning a fixed protocol but variable retrievedwealth. Both cases and their connections are consideredin the next section. III. PDF EVOLUTION AND STEADY STATESOLUTION
As previously shown [12], in the absence of regulation( χ = 0), Eq. (1) leads to progressive condensation to-wards w = 0, and a heavy tail for large w develops, with-out actually reaching a normalizable steady state. The inclusion of the drift with χ > w = 1, without loss of generality, since it is equivalentto performing the change of variables w/ ¯ w → w .For both families of kernels g ( w ), we develop inSecs. III A and III B, the setting I: fixing β = 1 and ad-justing γ selfconsistently. Setting II (adjustable β and γ = 1) is discussed in Sec. III C, where we show how thesteady states in both settings are related.Finally, the wealth PDFs are characterized in terms ofa widely used inequality indicator, the Gini coefficient,which can be estimated as [15], w . . . . . P ( w , t ) (a) χ = 0 . w = 0 .
50 1 2 3 4 w . . . . . P ( w , t ) (b) χ = 0 . w = 1 .
00 1 2 3 4 w . . . . . P ( w , t ) (c) χ = 1 . w = 1 . FIG. 1:
Evolution of the wealth PDF with piecewise-linear drift , for values of χ and w indicated in the legends,at times increasing (from lighter to darker) from t = 0 to t = 2 at each ∆ t = 0 .
1, and from t = 2 to t = 5 at each∆ t = 0 .
5. The solution of the stationary FPE is also plotted(green dashed line). G ( t ) = 1 − w (cid:90) ∞ xP ( x, t ) A ( x, t ) dx . (8) A. Piecewise-linear tax with exemption limit w The evolution of P ( w, t ), under the taxation ruled bythe piecewise linear kernel defined in Eq. (6), is shownin Fig. 1 for different values of χ and w , starting froman initial condition that results from the driftless evolu-tion, at a time where the Gini index is G (cid:39) .
59. Inall cases, increasing χ makes the final state less spread.The condensation at w = 0 is suppressed, meaning theabsence of a majority at extreme poverty, and the heavytail for large w becomes restricted by a cutoff, reducinglarge fortunes and oligarchy. In Fig. 1a, for w = 0 .
5, wefind a picture qualitatively similar to that of the linearcase [12], where the PDF has a positive skewness. Rais-ing the threshold w , the PDF can become almost flat w . . . . P s ( w ) − − − − (a) χ = 0 .
20 1 2 3 4 5 w . . . P s ( w ) − − − − (b) χ = 0 . w P s ( w ) − − − − (c) χ = 4 . FIG. 2:
Stationary wealth PDF, with piecewise-lineardrift , for fixed χ according to the legend, and different valuesof w varying (from light to dark lines) at each ∆ w = 0 . w (cid:39) (see Fig. 1b), meaning uniformly populated wealth lay-ers. For even larger w , the skewness is inverted at longtimes, with a mode larger than the average value ¯ w = 1(e.g., Fig. 1c). These are new features introduced by thepiecewise function. The evolution of the Gini index andfurther details are shown in Appendix A.The stationary PDFs for fixed χ and a large set of val-ues of w are shown in Fig. 2. Increasing the rate χ nar-rows the PDF around the mean, particularly, the cut-offat low w is shifted such that wider ranges of poverty areeliminated. This behavior is also observed in the linearcase and is not noticeably affected by w [15]. The ex-emption threshold w affects more strongly the interme-diate and large- w layers. The PDF can become bimodal(e.g, Fig. 2b), indicating classes with defined asset level,but scales are not well separated. The change of skewe-ness can be observed in these plots for not too small χ (Figs. 2b-c). Moreover, in general, raising the threshold w leads to a more effective cut-off such that assets thatsurpass that level tend to be suppressed. This is due tothe flow of wealth above w towards the population withlower assets, depopulating the large- w tail.The long-time value G s of the Gini index is shown fordifferent values of χ in Fig. 3a. It decreases by raising w ,as expected, however, a finite minimal level is attainedat a limiting value of w . The larger is χ , the moresensitive is G s to w . We have considered the full rangeof values for completeness, although some intervals ofthe parameters may be unrealistic. For instance, at thelimiting value of w , the slope γ s becomes divergent, tokeep the average ¯ w fixed (see Appendix A), and the PDF . . . . . . w . . . . G s χ = 0 . χ = 0 . χ = 0 . χ = 0 . χ = 1 . χ = 2 . χ = 4 . (a)0 . . . . . . w . . . . . . w m a x (b) FIG. 3: (a)
Stationary Gini index for different values of χ as a function of w . (b) Mode of the wealth PDF (filledsymbols) vs. w . The second maximum, when it exists, isalso plotted (hollow symbols). Lines are a guide to the eye. becomes truncated as can be observed in Fig. 2.In Fig. 3b, we plot the mode, w max (filled symbols), aswell as the second maximum (hollow symbols) wheneverit exists. For low values of χ , the mode is weakly sensitiveto w and remains below ¯ w (e.g., case χ = 0 . χ , the mode isshifted towards the line w max = w , and can exceed themean value ¯ w (unity, in our examples). When a secondpeak at larger w develops (which occurs for not too large χ ), it can become the mode as w increases (e.g., for χ = 0 . B. Power-law taxation
Stationary PDFs for different values of α and χ are ex-hibited in Fig. 4. As in the piecewise-linear case, there isa narrowing of the PDF with increasing χ . Flat, bimodal,and negatively skewed PDFs can also emerge when α > w P s ( w ) − − − − (a) χ = 0 .
50 1 2 3 4 5 w P s ( w ) − − − − (b) χ = 4 . FIG. 4:
Stationary wealth PDF, for power-law drift , P s ( w ), for different values of α at each ∆ α = 0 . α ∈ [0 . , . α = 1 for α ∈ [1 , χ accord-ing to the legend. Inset: same data in log-log scale to showthe cutoff. The stationary values of index G vs. α , for fixed valuesof χ , are presented in Fig. 5a. When comparing thesecurves with the respective ones of the piecewise-linearcase in Fig. 3a, we notice a matching for α (cid:39) w (cid:38)
1. While the progressive taxation with α > w , we find themain differences for α <
1, due to its regressive character.In the limit α →
0, due to the requirement of wealth α . . . . G s χ = 0 . χ = 0 . χ = 0 . χ = 0 . χ = 0 . χ = 4 . (a)0 5 10 15 20 α . . . . w m a x (b) FIG. 5: (a)
Stationary Gini index for different values of χ as a function of α . (b) Mode of the stationary wealthPDF (filled symbols) vs. α . The second maximum, when itexists, is also plotted (hollow symbols). Full lines are a guideto the eye. conservation, from Eq. (1), we have (cid:90) ∞ wP ( w, t ) dw = γ (cid:90) ∞ P ( w, t ) dw , (9)yielding constant γ = ¯ w . Then f ( w ) = χ ( ¯ w − γ ) = 0 ⇒ ∂∂w [ f P ] = 0 , (10)implying that the FPE drift term goes to zero when α →
0. Therefore, the dynamics evolves towards condensationand long tails. Even though, the sublinear kernel doesproduce relatively low values of the Gini index for largeenough χ . In fact, G s rapidly decreases with α in theregressive case. With regard to the maxima shown inFig. 5, the mode overcomes the average ¯ w and a secondmaximum can appear only for very large values of theexponent α . C. Setting II
Up to now, we developed the setting I, fixing theamount of wealth shared (by fixing β = 1) and adaptingthe coefficient γ ( t ) to conserve the average wealth. Now,we analyze setting II, where γ = 1 and β is adjusted.The stationary solutions can be related by identifyingeach drift terms in Eq. (4), yielding β IIs = 1 /γ Is χ II = χ I γ Is . (11) w . . . . . P ( w , t ) (a) χ = 2 . w = 1 . γ = 1 . t . . . . G χ = 0 . w = 1 . β = 1 . χ = 2 . w = 1 . γ = 1 . (b) FIG. 6:
Setting II: evolution for piece-wise lineardrift. (a) Wealth PDF (a), P ( w, t ) vs. w , with variable β , for the same times used in Fig. 1. (b) Gini index vs. timefor settings I (black) and II (light green), with parametersverifying Eqs. (11) leading to the same steady state). As an illustration, we followed the evolution of case w P s ( w ) − − − − (a) χ = 1 .
00 1 2 3 4 w P s ( w ) − − − − (b) χ = 2 . FIG. 7:
Setting II: stationary wealth PDF withpiecewise-linear drift , for fixed χ according to the legend,and different values of w varying (from light to dark lines)at each ∆ w = 0 . II, with parameters that verify the above relations withrespect to setting I in Fig. 1b. In fact, the steady statecoincide (compare Figs. 1b and 6a). However, the tempo-ral evolution differs, as can be also observed by followingthe Gini indexes over time, in Fig. 6b. The steady stateis reached faster within setting II.In contrast to setting I, where increasing the thresh-old w reduces the inequality measured by G s , in settingII, a different dependence on w emerges, as shown inFig. 6b. The stationary value of the Gini index is ratherinsensitive to w and even increases with w . A highthreshold exempts people with lower level of wealth butthe collected wealth is smaller. Then it is better to re-move the exemption. The corresponding steady PDFsare presented in Fig. 7. For the power-law taxation, theGini index decreases with α (not shown).Of course, in the limit w → α →
1, yielding thelinear case, both setting coincide. . . . . . . w . . . . . . G s χ = 0 . χ = 1 χ = 2 χ = 10 FIG. 8:
Stationary Gini index for setting II as a func-tion of w . Lines are a guide to the eye. IV. FINAL REMARKS
Regulatory processes seem to be necessary to coun-teract concentration trends. We studied nonlinear re-distributive mechanisms that allow to reduce inequality.Two families of taxation kernels were studied: piecewiselinear, with an exempting threshold, and power law. Weperformed numerical integration of the FPE to follow thetime evolution of P ( w, t ), starting from a given initialunequal distribution and also the stationary PDF P s ( w )was directly obtained.These studied tax protocols produce economic mobil-ity against the natural trend towards the condensation at w = 0 that occurs in the regulation-free case. The nonlin-earity brings new features with respect to the linear case,allowing to achieve greater social equality. Moreover, dis-tributions with peculiar stylized facts can emerge. Formoderate values of the control parameters, distributionscan be bimodal, which indicates the stratification andcoexistence of the population in economic classes withdistinct characteristic asset levels. A flat profile can alsoemerge, and, for strong regulation, the skewness of thewealth PDF changes sign. We also discussed the similar-ities and differences between two settings, where eitherthe collected or the divided wealth coefficients are ad-justed. Depending on the adjustment made, a higherexemption threshold w can be detrimental for equality.Let us also remark that the results can be translatedto those of arbitrary ¯ w by a simple scaling. In fact,from Eq. (1), we have P ( w, t ; ¯ w ) = P ( w/ ¯ w, t ; 1) / ¯ w , to-gether with the identifications: w (1) = w ( ¯ w ) / ¯ w (and γ unchanged) in the piecewise-linear case, and γ (1) = γ ( ¯ w ) ¯ w α − in the power-law case.As a perspective for future work, it would be inter- esting to study the underlying agent exchange dynamics,the effect of redistribution applied at discrete time steps,the combination of a power-law kernel with exemptionthreshold w , and also the impact of the spatial dimen-sion [18] going beyond the mean-field approximation.We acknowledge partial financial support by the Co-ordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Su-perior - Brazil (CAPES) - Finance Code 001. C.A. alsoacknowledges partial support by Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). Wethank Bruce Boghosian for elucidating methods used inhis previous works. [1] A. Dragulescu, V.M. Yakovenko, Statistical mechanics ofmoney, Eur. Phys. J. B 17, 723 (2000).[2] M. Patriarca, E. Heinsalu, A. Chakraborti, Basic kineticwealth-exchange models: common features and openproblems, Eur. Phys. J. B 73, 145 (2010).[3] Econophysics of Wealth Distributions: Econophys-Kolkata I ed. by A. Chatterjee, S. Yarlagadda, B.K.Chakrabarti. Springer Science & Business Media (2005).[4] C.F. Moukarzel, S. Goncalves, J.R. Iglesias, M.Rodrguez-Achach, and R. Huerta-Quintanilla, Wealthcondensation in a multiplicative random asset exchangemodel, EPJ Special Topics 143, 75 (2007).[5] J. Li, B. Boghosian, C. Li, The Affine Wealth Model:An agent-based model of asset exchange that allowsfor negative-wealth agents and its empirical validation,Physica A 516, 423 (2019).[6] https://en.wikipedia.org/wiki/List_of_countries_by_wealth_equality [7] B.M. Boghosian, Is Inequality Inevitable?, ScientificAmerican, Nov. 1 (2019).[8] T. Piketty, Le Capital Au Xxie Sicle (ditions du Seuil,2013).[9] Z. Burda, P. Wojcieszak, K. Zuchniak, Dynamics ofwealth inequality, C. R. Physique 20, 349363 (2019)[10] B.-H. F. Cardoso, S. Goncalves, and J. R. Iglesias,Wealth distribution models with regulations: dynamicsand equilibria, Physica A 124201 (2020).[11] C. Chorro, A simple probabilistic approach of the Yard-Sale model, Statistics and Probability Letters 112, 35(2016).[12] B. Boghosian, Kinetics of wealth and the Pareto law,Phys. Rev. E 89, 042804 (2014).[13] B. Boghosian, FokkerPlanck description of wealth dy-namics and the origin of Pareto’s law, IJMP C 25,1441008 (2014).[14] B. Hayes, Follow the money, Am. Sci. 90, 400 (2002).[15] B. Boghosian, A. Devitt-Lee, M. Johnson, J. Li, J.A.Marcq, H. Wang, Oligarchy as a phase transition: Theeffect of wealth-attained advantage in a FokkerPlanck de-scription of asset exchange, Physica A 476, 15 (2017).[16] J. Lie, B. Boghosian, Duality in an asset exchange modelfor wealth distribution describes a very interesting dual-ity symmetry of this Fokker-Planck equation, Physica A497, 154 (2018).[17] B. M. Boghosian, A. Devitt-Lee, H. Wang, Describingrealistic wealth distributions with the extended Yard-Sale model of asset exchange, arXiv:1604.02370v1 (2016).[18] J. Novotn´y, On the measurement of regional inequal-ity: does spatial dimension of income inequality matter?,Ann. Reg. Sci. , 563580 (2007). Appendix A: Numerical integration of the FPE
In order to obtain the solution of Eq. (1) numerically,we performed the change of variables w = − ln(1 − y ) , (A1)that univocally maps the interval [0 , + ∞ ) into the inter-val [0 ,
1] [15]. With this change, Eqs. (2)-(5) become A ( y, t ) = (cid:90) ∞ y P ( w ( x ) , t )1 − x dx , (A2) B ( y, t ) = (cid:90) y ln (1 − x )1 − x P ( w ( x ) , t ) dx , (A3) f ( y, t ) = χ (cid:16) β ¯ w − γg ( w ( y )) (cid:17) , (A4)¯ w = − (cid:90) ln(1 − y )(1 − y ) P ( w ( y ) , t ) dy . (A5)Additionally, the normalization condition, meaning con-servation of the population size, becomes (cid:90) P ( w ( y ) , t )1 − y dy = 1 . (A6)Then, Eq. (1) can be rewritten as ∂P∂t + (1 − y ) ∂C∂y = (1 − y ) (cid:26) (1 − y ) ∂ D∂y − ∂D∂y (cid:27) , (A7)where C and D are C ≡ C ( y, t ) = f ( y, t ) P , (A8) D ≡ D ( y, t ) = 12 (cid:0) ln (1 − y ) A + B (cid:1) P . (A9)Equation (A7) was integrated using a standardforward-time centered-space algorithm.In setting I, β = 1 and γ evolves satisfying γ ( t ) (cid:90) ∞ g ( w ) P ( w, t ) dw = ¯ w , (A10)which means conservation of the total wealth owned bythe individuals (or average wealth per individual, sincethe population size is conserved).In setting II, γ = 1 and β is ruled by β ( t ) ¯ w = (cid:90) ∞ g ( w ) P ( w, t ) dw . (A11)For the piecewise-linear case, the time evolution of γ within setting I is shown in Fig. 3a, and that of theGini index in Fig. 3b. In the absence of regulation( χ = 0), condensation would proceed, leading to dra-matic inequality, with the Gini index monotonically in-creasing. Differently, for χ >
0, the Gini index stabilizes.Depending on the initial condition, this stabilization canoccur from below (e.g., for χ = 0 . χ . While the initial rate of de-crease depends on χ , the final stabilization value is ruledby w too. Of course, for an initial PDF with sharp cutoffbelow w , the drift with be ineffective. t γ χ = 0 . w = 0 . χ = 0 . w = 0 . χ = 0 . w = 0 . χ = 0 . w = 0 . χ = 0 . w = 1 . χ = 1 . w = 0 . χ = 1 . w = 1 . (a)0 1 2 3 4 t . . . . . G (b) FIG. 9: (a) Time evolution of γ , from the numerical inte-gration of FPE (1), for the values of χ and w of Fig. 9. Thelong-time value of γ is in good agreement with that obtainedby direct integration of the stationary FPE. (b) Evolution ofthe Gini index G ( t ). Dotted lines indicate the steady statevalues. The computational cost increases when reaching thedrift-less limit χ = 0. Appendix B: Direct integration of the steady FPE
In the particular case of the linear rule g ( w ) = w ,recovered for w = 0 or α = 1, we have β = γ = 1,independently of the value that χ > γ s =1 (or β s = 1).Otherwise, the stationary value γ s was obtained self-consistently from the integration of the stationary formof the FPE (1), which for no flux boundary conditionsreads 12 µ (cid:48) − f P = 0 , (B1)where µ ≡ µ ( w ) = ( w A + B ) P ( w ) and “ (cid:48) ” mean differ-entiation with respect to w .In order to solve this equation, we generalized the nu-merical procedure described in Ref. [15]. It consists insplitting Eq. (B1) into the following coupled linear dif-ferential equations, namely, A (cid:48) = − P = − µ/ ( w A + B ) , (B2) B (cid:48) = w P = w µ/ ( w A + B ) , (B3) µ (cid:48) = = 2 f P = 2 f µ/ ( w A + B ) , (B4)with the initial conditions A (0) = 1, B (0) = µ (0) =0, together with the normalization condition, namely, A = lim w →∞ A ( w ) = 0. This integration however isnot straightforward due to the singular behavior of P ( w ) . . . . . . w γ s χ = 0 . χ = 0 . χ = 1 . χ = 2 . χ = 4 . (a)0 2 4 6 8 10 α − − − γ s χ = 0 . χ = 0 . χ = 1 . χ = 2 . χ = 4 . . . . . . . . . (b) FIG. 10: Stationary value of γ as a function of w (a) and α (b), for different values of χ indicated in the legend. In (a),the vertical dotted lines indicate the asymptote. In (b), theinset is a zoom of the vicinity of the origin. Full lines are aguide to the eye. near the origin, that behaves as P ( w ) (cid:39) Cw exp (cid:16) (cid:90) w f ( x ) x dx (cid:17) , (B5)where C is a constant. Then, we use the final valuein the interval (0 , δw ), with δw (cid:28) C , whichmust be determined from the normalization constraint A = lim w →∞ A ( w ) = 0. From the plot of A ( C ) vs. C , using a Newton-Raphson (NR) procedure, C can bedetermined by solving A ( C ) = 0, which has a single root.This is, essentially, the procedure described before for thelinear case [15]. In the nonlinear case, we must still de-termine the value of γ s (in setting I) or β s (in setting II)that defines f ( w ), under the constraint of conservationof the average wealth ¯ w . Then, a second NR procedureis required to find the correct value of γ s or β s for eachvalue of C that enters in the first NR scheme. (Actu-ally in numerical integration, we also use the change ofvariables given by Eq. (A1).) The steady solutions P s ( w )found through this procedure are in accurate agreement with the long-time solutions obtained by numerical inte-gration of the time-dependent FPE, as illustrated in theinsets of Figs. 1 and 7.For fixed β = 1 (setting I), the stationary value of γ s isplotted in Fig. 10, for given values of χ , as a function of w (a) and α (b). In case (a), γ s increases from unity at w = 0 (pure linear case) diverging at a finite value w c (indicated by dotted vertical lines), following the scaling γ s ∼ ( w c − w ) − . This divergence implies a trunca-tion of the PDF at the critical value w c , as observed inFig. 4. For the power-law rule, the coefficient γ playsa different role. The curves first increase from γ s = 1,up to a maximal value, and then decay exponentially to-wards zero. Notice that when, α = 1 (pure linear case), γ s = 1. When α = 0, from Eq. (10), we have γ = ¯ ww