Inequality, a scourge of the XXI century
José Roberto Iglesias, Ben-Hur Francisco Cardoso, Sebastián Gonçalves
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Inequality, a scourge of the XXI century
Jos´e Roberto Iglesias ∗ Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, BrazilEscola de Gest˜ao e Neg´ocios, Programa de P´os-Gradua¸c˜ao em Economia,UNISINOS, Porto Alegre, RS, Brazil andInstituto Nacional de Ciˆencia e Tecnologia de Sistemas Complexos,INCT-SC, CBPF, Rio de Janeiro, RJ, Brazil
Ben-Hur Francisco Cardoso † Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
Sebasti´an Gon¸calves ‡ Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil andURPP Social Networks, University of Z¨urich,Andreasstrasse 15, CH-8050 Z¨urich, Switzerland (Dated: May 14, 2020) bstract Social and economic inequality is a plague of the XXI Century. It is continuously widening,as the wealth of a relatively small group increases and, therefore, the rest of the world shares ashrinking fraction of resources. As an example, in 2016, the wealthiest 1% of US citizens possessed40% of the nation’s wealth, while in 2007 they had less than 35% [1] —considering the world’spopulation, the richest 1% have today 50% of the total wealth. This situation has been predictedand denounced by economists and econophysicists. The latter ones have widely used models ofmarket dynamics which consider that wealth distribution is the result of wealth exchanges amongeconomic agents. A simple analogy relates the wealth in a society with the kinetic energy of themolecules in a gas, and the trade between agents to the energy exchange between the moleculesduring collisions. However, while in physical systems, thanks to the equipartition of energy, the gaseventually arrives at an equilibrium state, in many exchange models the economic system neverequilibrates. Instead, it moves toward a “condensed” state, where one or a few agents concentrateall the wealth of the society and the rest of agents shares zero or a very small fraction of the totalwealth. According to Piketty [2], many American countries, but also some European ones, arefollowing this path. Here we discuss two ways of avoiding the “condensed” state. On one hand,we consider a regulatory policy that favors the poorest agent in the exchanges, thus increasing theprobability that the wealth goes from the richest to the poorest agent. On the other hand, westudy a tax system and its effects on wealth distribution. We compare the redistribution processesand conclude that complete control of the inequalities can be attained with simple regulations orinterventions. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Empirical studies of the distribution of income of workers, companies and countries werefirst presented, more than a century ago, by Italian economist Vilfredo Pareto. He assertedthat in different European countries and times the distribution of income follows a power lawbehavior, i.e. the cumulative distribution P ( w ) of agents whose income is at least w is givenby P ( w ) ∝ w − α [3]. Non-Gaussian distributions are denominated Levy distributions [4],thus this power law distribution is nowadays known as Pareto-Levy Distribution and theexponent α > / annual income (BRL) − − − c u m u l a t i v e f r a c t i o n o f t a x p a y e r s exponential power law α = 1 . FIG. 1. Cumulative fraction of taxpayers as a function of annual fiscal income, in Brazilian Real(BRL). Data collected from IRPF 2015 [10]. GDP (BRL) − − − − c u m u l a t i v e f r a c t i o n o f c i t i e s power law α = 1 . FIG. 2. Cumulative fraction of cities as a function of annual fiscal Gros Domestic Product, inBrazilian Real (BRL). Data collected from IBGE 2015 [11]. i is characterized by a wealth w i and a risk-aversion factor β i . During thewealth exchange between agents i and j , assuming that i wins, we have w ∗ i = w i + dw and w ∗ j = w j − dw, where w ∗ i ( j ) is the wealth of the agent i ( j ) after the exchange. There are different waysof defining the quantity dw transferred from the loser to the winner, but mostly two areconsidered: the fair (or yard-sale ) rule and the loser rule [12, 22, 23]. The first one statesthat dw = min[(1 − β i ) w i ( t ); (1 − β j ) w j ( t )], while in the second we have dw = (1 − β j ) w j ( t ).The first rule is called fair because the amount of wealth exchanged is the minimum of thequantities put at stake by the two agents and it is the same regardless of who wins, so therichest agent accepts to risk part of its wealth [22]. In the loser rule, on the other hand, thewinner receives the value risked by the loser; thus, as the richest agent most probably putsa larger amount at stake, this kind of exchange is only likely in situations where agents donot know the wealth of the others [15].Numerical [12, 20] and analytical [24] results with the fair rule model, or some variationsof it, point out to the condensation fate, i.e. a continuous concentration of all availablewealth in just one or a few agents, leading to an absorbing state where no more wealth isexchanged, a kind of “thermal death” of trade [25]. These results seem to describe somehowthe present path of the world economy, that experiences a continuous grow of inequality [2].Different modifications have been introduced in the models to overcome condensation.For example, increasing the probability of favoring the poorest agent in a transaction [20, 21],or introducing a taxation mechanism [13, 26], where periodically all agents pay taxes andthe amount collected is then in someway divided among the agents. Here, we study andgeneralize these two approaches. 5n Section II we study the former approach, using a rule suggested by Scafetta [12, 21,27, 28], where, in the exchange between the agents i and j , the probability of favoring thepoorest partner is given by: p = 12 + f × | w i ( t ) − w j ( t ) | w i ( t ) + w j ( t ) , (1)where f is a factor that we call social protection factor, which goes from 0 (equal probabilityfor both agents) to 1 / p of earn a quantity dw , whereas the richest one hasprobability 1 − p . It is evident that the higher the difference of wealth in a given pair ofagents, the higher the influence of f in the probability. For that reason we think f could bean indicator of the degree of government regulation or control of the economy of a country.However, although with this approach we avoid condensation, the factor f , in Eq. 1, cannot be transferred easily to quantitative economic measures in real scenarios. Besides, it cannot account for the low values of the Gini index of some countries such as the Scandinaviancountries, for example. Then, in Section III we study how different types of tax collectionand redistribution affect the distribution of wealth.It is important to mention that hereafter wealth ( w i ) and risk aversion factor ( β i ) aredistributed uniformly in the interval [0 ,
1] as initial conditions. While w i ’s evolve in time aspart of the dynamics, the values of β i ’s are fixed. A. Inequality
To measure inequality, we use three indicators: the share of the wealth of the wealthiest1% of the agents, the share of the wealth of the richest 10% of the agents (or, reciprocally,the share of wealth in hands of the poorest 90% of the agents), and the Gini index.The Gini index is a measure created in 1912 by Italian statistician Conrado Gini; it isusually used to measure the inequality of a distribution of income or wealth. It is calculatedas a ratio of the areas in the Lorenz curve diagram (see Fig. 5); if the area between theline of perfect equality and the Lorenz curve is a , and the area below the Lorenz curve is b ,then the Gini coefficient is a/ ( a + b ) [29]. In an operational way we evaluate the Gini indexas [30]: G ( t ) = 12 P i,j | w i ( t ) − w j ( t ) | N P i w i . (2)6he Gini index varies between 0, which corresponds to perfect equality ( i.e. everyone hasthe same wealth), and 1, that corresponds to the extreme inequality where only one agentpossesses all the wealth. II. SOCIAL PROTECTION . . . . . f . . . . . G without ZWAwith ZWA FIG. 3. Gini index at equilibrium as a function of the social protection factor, f ; symbols arecalculated with N = 10 and interpolating curves, with values of f separated by 0 .
01, with N = 10 . ZWA stands for zero wealth agents.
We show in Fig. 3 the Gini index at equilibrium as a function of the factor f [23]. Clearly,the lower the social protection factor, the greater the inequality until reaching the extremecase f = 0, where the system condenses ( G = 1). A significant contribution to the highinequality for low values of f is the large fraction of agents with zero wealth, which isgreater than 60% for f = 0 .
1. This is evident looking at the comparatively lower Gini curvewhen zero wealth agents are excluded. It is worth to mention that all results are averagesover 10 samples, and three different sizes of the system: N equal to 10 , and 10 . As The global per-capita wealth in 2014 was less than 10 USD [31], while in our artificial system the meanwealth is 0 .
5. Since USD unit is discretized by the minimum value of 0 .
01 USD, by analogy we set theminimum unit as 10 − . Therefore, we consider the wealth of an agent i equal to zero if w i < − .
965 1980 1995 2010 year . . . s h a r e o f w e a l t h top bottom . . . . . f . . . s h a r e o f w e a l t h top bottom FIG. 4. Fraction of wealth in hands of the top 1% and bottom 90%. Left: Evolution in the UnitedStates; data collected from Wolff [1]. Right: results for the fair rule in terms of the protectionparameter f ; symbols are calculated with N = 10 and interpolating curves, with values of f separated by 0 .
01, with N = 10 . Noticed the qualitative parallelism in the two figures for f < . f protection factor. the obtained results are almost independent of the size of the system, we have plotted justthe outcome for N = 10 and N = 10 .A similar relation between social protection and inequality is observed in Fig 4 (right),were we show the share of wealth of the upper 1% and the bottom 90% of the agents.Empirically, Fig. 4 (left) shows the evolution of the concentration of wealth in the hands ofthe upper 1% of the US society during the last fifty years, accompanied by the continuousdeclining of the wealth of the bottom 90%. We can interpret the growth of US inequality asthe decrease in time of some effective f factor.As complementary information, we represent in Fig. 5 the distribution of wealth bydrawing the Lorenz curves for some selected values of f . One point in a Lorenz curvetells us the fraction of total wealth that a fraction of the population possess. The effect ofreducing the protection factor f is very clear, producing curves that depart from the perfectequality line to the almost perfect inequality line in the extreme f = 0 case.8 .
25 0 . .
75 1 F . . . L p e r f ec t e q u a li t y f = 0 . f = 0 . f = 0 . f = 0 FIG. 5. Lorenz curves for the fair rule with N = 10 agents and for different values of the parameter f . L is the fraction of wealth that the fraction F of agents has. III. TAXES
In this section, we present results with a simple flat tax system on the wealth. In thesimulation, the tax collection works as follows: at each Monte Carlo Step, all agents pay afraction λ of its wealth as taxes . Then, the total amount collected is redistributed amongthe agents, according to two different criteria: universal and targeted. Different kinds oftaxes have been studied in ref. [32]. A. Universal Redistribution
Universal redistribution is the simplest case, where the total amount collected as taxesis equally distributed among all agents, irrespectively of their wealth. Similar taxationmechanism have already been proposed [13, 26], but assuming that β values are very closeto 1, and in the small transaction limit approximation. Despite this difference, our resultsare qualitatively analogous.We show in Fig. 6 and Fig. 7, respectively, the Gini index and the share of wealth ofthe top 1% and top 10% of the population as a function of λ , the tax percentage. Wecan observe that the higher the tax rate, the lower the inequality, as expected. Differently The wealth tax is equivalent to taxes on property or fortune, much less widespread than the income tax.However, we choose the former system because it is much effective in reverting inequality. f , the taxation mechanism can greatly reduce inequality.Moreover, by means of the tax index λ , it is possible to obtain all the possible values of theGini index, from the maximum inequality when there are no taxes ( λ = 0) to the trivialcomplete egalitarian society G = 0, if λ = 1. However, in real societies one does not expectthe value of λ to go beyond 0 . .
25 0 . .
75 1 λ . . . . G FIG. 6. Equilibrium Gini index as a function of λ , the tax index on fortune. Lines correspond tosimulations with N = 10 agents and symbols, with N = 10 . B. Targeted Redistribution
In the targeted case, the total amount collected as taxes is distributed among the p poorestfraction of the population (the targeted population). The universal case is recovered when p = 1. We show in Fig. 8 the Gini index as a function of both λ and p . We remark that, byrestricting the allocations to less that 1% of the people ( p ≤ − ), as it is the case in mostof governmental projets to help unemployed and/or very poor people, the effect, in termsof Gini coefficient, is almost unperceptible. Instead, it is necessary to expand assistance toat least 10% of the poorest people to achieve a real effect of reducing inequality. The figurealso shows that there is an optimum value of p = p ∗ that minimize the inequality for each10 .
25 0 . .
75 1 λ − − s h a r e o f w e a l t h top top FIG. 7. Share of wealth of the top 1% and 10% of the population as a function of λ , the tax indexon fortune. Lines correspond to simulations with N = 10 agents and symbols, with N = 10 . value of the tax rate λ , so there is an interesting non-trivial relation between λ and p in thiscase. − − λ − − p . . . . . . . . . . . . G FIG. 8. Equilibrium Gini index as a function of λ and p (the bottom fraction of agents). Resultsfor N = 10 agents. λ , for p = 1 (universalcase) and p = p ∗ (optimal targeted case). We remark that, for intermediary values of λ , particularly for λ ≈ .
35 the regulatory mechanism of helping just a fraction of thepopulation is more efficient to strongly reduce inequality. .
25 0 . .
75 1 λ . . . G p = 1 p = p ∗ FIG. 9. Equilibrium Gini index as a function of λ (tax index on wealth), for p = 1 (universal case)and p = p ∗ (optimal targeted case). Results for N = 10 agents. IV. DISCUSSION AND CONCLUSION
Economic inequalities are continuously increasing worldwide, with the exception of a fewcountries. It is a plague of the past and present centuries that threatens the economic sus-tainability all around the world. Inequalities arise in a very similar way to the condensationtendency of the models reviewed here. The ideal free market without regulations, whichseems to be fair because it gives everyone the same opportunities, it is not. It is an illusionthat fails because of the multiplicative factor behind, which increases the wealth of the firstfavored by random fluctuations, while the rest of the population goes irremediably to break-down. We have shown that even strong regulations, while avoiding condensation, are notenough to lead to a low Gini index. On the other hand, we have demonstrated that taxesare a possible way to get to any level of inequality, from the highest to the utopic minimum.12 practical evidence is the case of Scandinavian countries (where the Gini coefficient is ofthe order of G = 0 .
25) which exhibit both high taxes and low inequality.The model here presented is certainly a very simplified one; for example, taxes do notplay just a redistributive objective, but also are used by governments to finance public ad-ministration and to assure the infrastructure needed to economic development. Nonetheless,in spite on its simplicity the present model describes a very strong redistribution mechanism,that coincides with some public policies, like the “bolsa-familia” (family fellowship, allocateto very poor family groups in Brazil), or the small loans to jobless people in order to developtheir own business. Even at global level, and with taxes proportional to the possessions ofeach agent, the contribution of each one could be small but results in a low Gini coefficient,even lower than the one observed in the most egalitarian countries, like Denmark or Japan. Is has to be also noticed that the universal redistribution tax system is much easier toadministrate than a targeted redistribution system. Even if the tax on fortune seems a veryreasonable mechanism to reduce inequalities, we are presently working in the effect of otherkind of taxes, like tax on the profits, revenues and consumption. Also, more sofisticatedredistributive policies are also investigated. Results will be communicated soon.We conclude that the yard-sale model, in spite of its simplicity, is able to reproduce someproperties of modern economies, particularly the tendency to continuously increase inequal-ity in the absence of regulations. We have also demonstrated that an effective redestributivemechanism, provided by a policy of taxing great fortunes and incomes, can greatly reduceinequality.
ACKNOWLEDGEMENTS
This study was partially financed by the Coordena¸c˜ao de Aperfei¸coamento de Pessoalde N´ıvel Superior - Brasil (CAPES) - Finance Code 001. BFC acknowledges Brazilianagency Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for scholar-ship. JRI acknowledges Brazilian agency Conselho Nacional de Desenvolvimento Cient´ıficoe Tecnol´ogico (CNPq) for support. SG acknowledges CAPES for support, under fellow-ship [1] E. N. Wolff,
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