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taxation model: Investors in groups
Physica A 537 (2020) 122588
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Investment/taxation model: Investors in groups P.M.C. de Oliveira
∗
Instituto de Física, Universidade Federal Fluminense, Niterói - RJ, 24210-340, Brazil Universidade Federal da Integração Latino Americana, Foz do Iguaçu - PR, 85867-000, Brazil National Institute of Science and Technology - Complex Systems, Brazil
article
info
Article history: Received 7 June 2019 Received in revised form 15 August 2019 Available online 16 September 2019
a b s t r a c t Individuals invest their capital and/or working skills, multiplying their wealths. This process is modeled by grouping investors according to different network possibilities. They also pay taxes according to progressive tax rates (increasing for increasing wealths) or regressive (the contrary). Independent of the particular rule of investment, there is a dynamically induced transition where all the population wealth falls in hands of a single individual for regressive taxation. In this case, the economic evolution is extinct. On the other hand, for progressive taxation the economic evolution continues forever. The critical properties of this transition, however, depend on the geometric arrangement adopted for investments. Here, the transition is studied in different spatial dimensions. For regular lattices a dimension universality is found, where the critical properties depend only on the lattice dimension. © 2019 Elsevier B.V. All rights reserved.
The main plot in Fig. 1, to be discussed later, shows what physicists call a critical quantity. Its importance, both theoretically as well as in numerous practical applications (a myriad of electronic devices for instance) resides in the possibility of controlling this quantity along the vertical axis by fine tuning its control parameter along the horizontal axis. Thanks to the tangency of the curve with the vertical axis, and the consequent mathematical singularity at the critical point. It is also the signature of continuous phase transitions (magnetic materials for instance), the behavior of the system abruptly changes when the critical point is crossed. Also, near the critical point, the behavior of these systems is governed by macroscopic concerns instead of details of its microscopic constituents, leading to the phenomenon of universality: completely different systems (for instance, an uniaxial ferromagnetic material near its magnetic transition temperature and a fluid like water near 374C, above which the coexistence of vapor and liquid phases no longer exists) present exactly the same critical exponent β for the respective quantities (the magnetization in the first case or the difference between liquid and vapor densities in the other). Due to this universality phenomenon, very simplified theoretical models sharing the same few macroscopic features with the complicated system under study can be applied with success in order to predict its macroscopic properties. For a review, see [1]. Although the theory of continuous phase transitions, critical phenomena and universality was well stablished for systems in thermodynamic equilibrium, its features were observed also in other out-of-equilibrium systems. Even in cases where thermodynamic concerns are absent, not only in the domain of Physics but also in many other domains like Biology, Ecology, Chemistry, Sociology, Economics, i.e. dynamic evolutionary systems in general. Nowadays, the strategy of applying very simplified dynamic models in order to study phenomena observed in these domains is widespread. In [2] the economic evolution of a society is modeled with two dynamic ingredients, repeated every year: (I) Individual agents invest their capital or work potential, altogether represented by their ‘‘wealths’’, Wn (n = 1, 2 . . . N); (II) They pay taxes. Step I is a simple multiplication of Wn by a random factor tossed from an arbitrary but fixed probability distribution. ∗ Correspondence to: Instituto de Física, Universidade Federal Fluminense, Niterói - RJ, 24210-340, Brazil. E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.122588 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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P.M.C. de Oliveira / Physica A 537 (2020) 122588
Fig. 1. Critical behavior: A tiny tuning of the control parameter p, near the critical point (here p = pc = 0), triggers a huge but continuous jump of the order parameter m. A phase transition occurs when the critical point is surpassed, from an inactive phase (m = 0 on the lefthand side) towards an active phase (m > 0 on the righthand side). The curve is tangent to the vertical axis at the critical point, its mathematical form is m ∼ pβ , with critical exponent β < 1 (here β = 0.64). The error bars are smaller than the symbols. Inset: The singularity at pc disappears when a uniform external field is applied (here, h = 0.1, 0.08, 0.04 and 0.025, smooth curves from top down).
In step II a tax rate depending linearly on the wealth Wn is adopted, defined by a fixed parameter p, the angular coefficient (slope) of the quoted linear dependence. For p > 0 the taxation is progressive, i.e. rich agents ∑ pay larger tax rates than poor agents. For p < 0 it is regressive, the contrary [3]. The wealth shares wn = Wn /S, where S = n Wn is the total population wealth, are considered. In particular, the largest share w1 corresponding to the richest agent. Also, these shares are always considered in decreasing order, so the index n does not correspond to one specific agent, but to its position along the current ranking of wealths. For each fixed p this dynamic rule is applied iteratively, year after year. For regressive taxation (p < 0), after some time the total population wealth is concentrated in the hands of a single agent, w1 = 1 with wn = 0 for all other agents n > 1. In this regressive case, the economic evolution of the society stops, it is what physicists call an absorbing state of the dynamics, biologists would say extinction. For progressive taxation (p > 0), the distribution of wealth shares fluctuates forever, but after some time their time averages over the last years reaches a steady state with wn < 1 for all agents. Therefore, there is a dynamically induced transition from activity towards inactivity (absorbing state) when one crosses the frontier p = pc = 0 from progressive to regressive taxation. The order parameter describing this transition is m = −⟨log w1 ⟩ where ⟨. . . ⟩ represents the time average in the steady state, as well as a further average over different realizations. The transition occurs always at the frontier p = pc = 0 separating progressive from regressive taxation, for numerous different rules of investment, step I. Fig. 1 shows the order parameter m as a function of the control parameter p for one such rules. In short, the model consists of the following steps, repeated each year. ∑ Step I (investment). The wealth shares w1 > w2 > w3 > · · · > wN are normalized, n wn = 1. Each one is multiplied by a random factor fn obtained from an arbitrary but fixed rule, Wn = fn wn
∑
and the new sum S = n Wn is calculated. Step II (taxes). Each agent pays taxes according to a linear tax rate Wn′ = Wn (1 − A − p Wn /S) where the extra parameter A plays no role, it is included only to assure the paid tax is positive and not larger ∑than the wealth, 0 < A < 1 and 0 < A + p < 1. Before starting the new year, the current wealth shares wn′ = Wn′ / i Wi′ are calculated in order to be used in the next year. This renormalization of wealths can be interpreted as a simple redefinition of the monetary unit (currency), always taken as the total population wealth at the beginning of each year.
P.M.C. de Oliveira / Physica A 537 (2020) 122588
3
The order parameter m = −⟨log w1 ⟩ has an interesting interpretation within the Economics point of view, related to social inequalities. The larger it is the smaller are these inequalities, i.e. the smaller is the ratio between the richest agent and the average wealth share 1/N. Within this interpretation, even a small degree of progressive taxation through a small positive value for the control parameter p is effective in mitigating inequalities, thanks to the critical behavior shown in Fig. 1 and its characteristic tangency between the curve and the vertical axis. Also, the smaller the critical exponent β < 1 the larger is this effectiveness. Besides progressive taxation, another traditional way to mitigate social inequalities is a redistribution of the paid taxes to all individuals. An extra step III can be introduced into the above rules, before the wealth distribution renormalization. Step III (redistribution). A fraction R of the total taxes
∑
Wn (A + p Wn /S)
n
paid by the population is uniformly redistributed among all agents. This ingredient plays the same role of a uniform external field applied to traditional thermodynamic systems, namely to turn critical, singular curves like the main plot in Fig. 1 into smooth curves as shown in the inset of the same Fig. 1. Here, the external field is measured by a further control parameter h defined below 1 h
= − log R
representing the degree of redistribution. The critical situation corresponds to h → 0, no redistribution. Following this trend, the smooth curves tend to the critical one [4]. This model inserts into a class of similar others, references [5] give good reviews and many further references. In particular, the pioneering work [6] worths to be a basis for comparison. The investment step I, a multiplicative random process, is the same. However, a particular probability (log normal) distribution for the factors fn was adopted in this work, whereas the current model relaxes this restriction. Indeed, any probability distribution for the investment factors fn produces a Pareto distribution of wealths among the population, provided it is the same for all agents. This point is formally demonstrated in [7], in agreement with the observed behavior of numerous real markets. Here, a similar universality is observed concerning the phase transition, always occurring at p = pc = 0. Also, the taxation step II takes into account only a uniform taxation rule in [6], i.e. only the parameter A is present (p = 0), therefore the transition could not be captured. Also the criticality could not be captured, since only the full redistribution (R = 1 or h → ∞) is taken into account. We tested also the inclusion of a pair-coupling Jij where agents i and j trade among themselves, like [6]. One such agent (payer) transfers a fraction of its wealth to the other (receiver). The overall result of this coupling is thus to transfer wealth from the richest agents to the poorest agents. The effect of this coupling is thus similar to the inclusion of the already quoted external field, collected taxes uniformly redistributed. Therefore, it is not considered in the numerical results shown here. The spatial correlations among agents induced by some network is present only in the investment grouping mechanism explained later. Concerning the control of social inequalities, within similar models, some references are also provided [8]. Among the numerous investment rules one can invent, every one leading to the dynamically induced transition at pc = 0, a simple one is to take half the agents, doubling their wealths (fn = 2) while the other half is untouched (fn = 1). One can interpret this as a random walk in the would-be ‘‘space’’ of N agents repeated until covering a number N /2 of them (each agent covered by the random walk has its wealth doubled only once, even when revisited). In [2] this half-half division would be taken at random, which corresponds to a random walk over an infinite-dimensional hyper-cubic lattice, a network of N sites every one linked to any other. The purpose of the current work is to study other possible networks, to investigate how the links forming groups of investors acting together would be reflected into the transition criticality. A first possibility is to consider the agents positioned along a 1-dimensional ring with N sites. This rule corresponds to Fig. 1. The same rule generates the steepest straight line in Fig. 2, no longer tangent to the vertical axis. It means that progressive taxation (Fig. 1) is more effective than redistribution (Fig. 2) when the purpose is to mitigate inequalities. An alternative is to place the agents on a 2-dimensional L × L square lattice (N = L2 ), tossing N /2 agents through a random walk starting from a random site, and doubling their wealths. The equivalent to Fig. 1 (not shown) is similar with a larger critical exponent. Another alternative is a triangular lattice (instead of square), also 2-dimensional. An interesting observation is the coincidence (within the error bars) of results on the square or triangular lattices. In particular, both share the same exponent β . Also as function of the external field, the middle straight line on Fig. 2 exhibits this coincidence. Again by analogy with traditional thermodynamic phase transitions, they seem to belong to the same ‘‘universality class’’ which would be determined only by the geometrical dimension of the network where the agents are located. This behavior implies that somehow to approach the critical situation p = h = 0 corresponds to increase the geometrical length scale relevant to the observed criticality. Following this trend, this length scale becomes much larger than the microscopic lattice details. In order to confirm this behavior, Fig. 3 shows the results for different 3-dimensional lattices. Indeed, the coincidence occurs again when the critical point (p = h = 0) is approached from both directions. A larger exponent β is shared by all 3-dimensional cases. In infinite dimension (the random case) the exponent is larger yet. In all these cases of regular lattices, plots like Fig. 2 result in asymptotically straight lines towards the origin, i.e. the mathematical form is m = ah1 , with decreasing slopes. Numerical results are displayed in Table 1.
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P.M.C. de Oliveira / Physica A 537 (2020) 122588
Fig. 2. Order parameter as a function of the external field at the critical point p = pc = 0. Top: The same one-dimensional ring as in Fig. 1. Middle: 2-dimensional square and triangular lattices (different colors on line). Bottom: Fractal cluster constructed over a square lattice by erasing random sites, at the percolation threshold concentration 0.59274621 of remainder sites.
Fig. 3. Order parameter as a function of the external field at the critical point p = pc = 0, for 3-dimensional lattices. Simple cubic (open circles, straight line), body-centered cubic (filled circles) and face-centered cubic (squares).
Worth to mention that even in the absence of the coupling Jij , like [6], the lattice geometry plays a fundamental role because the set of neighbor agents investing together, i.e. the whole set under the same investment random factor (step I), is defined within this geometry by the random walk. Regular lattices are obviously not realistic networks for groups of investors (clients of the same investment firm for stock market, or members of the same workers representative institution, for example). More realistic networks presenting fractal properties could be tested. However, this is not the purpose of the current work, but instead to explore the similarities of this non-equilibrium transition with the traditional behavior of thermodynamic equilibrium transitions, in particular the phenomenon of lattice dimension universality. Anyway, an example of non-regular, fractal network is
P.M.C. de Oliveira / Physica A 537 (2020) 122588
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Table 1 Critical exponent β of the order parameter m ∼ pβ without redistribution (h → 0) as in the main plot of Fig. 1, and slope a of the straight line m = ah at the critical point (p = pc = 0) as in Fig. 2, in 1, 2, 3 and ∞ dimensions. Error bars are smaller than 10−2 in all cases. 1D
2D
3D
∞D
β = 0.64 a = 7.20
0.80 1.35
0.84 1.18
0.85 1.16
tested: percolation clusters obtained by diluting the sites of a square lattice at the percolation threshold, i.e. a fraction 0.59274621 of present sites [9]. One example is shown in Fig. 2, bottom curve. In this case one can observe the tangency occurring between the curve and the horizontal axis (instead of vertical as in Fig. 1), meaning that the strategy of redistribution is very inefficient when the purpose is to mitigate inequalities. The mathematical form is m ∼ he with the exponent e > 1 (e ≈ 1.3 in Fig. 2, bottom curve). As a function of p, however, an intermediate behavior between that of 1 and 2 dimensions was found. Concluding, an investment/taxation model was introduced with investors acting together in groups. Each year, after the investment step, agents pay taxes according to a progressive or regressive taxation. This iterative dynamic process leads to a phase transition: for regressive taxation the dynamic evolution stops with the entire population wealth in hands of a single agent, whereas for progressive taxation the economy evolves forever with the population wealth distributed among all agents. The wealth share w1 corresponding to the richest agent, averaged in time and realizations, defines the order parameter m = −⟨log w1 ⟩ for this transition, with m = 0 for regressive taxation. For progressive taxation, corresponding to a positive control parameter p, the critical behavior of the transition can be determined. A second control parameter h related to a uniform redistribution of paid taxes among all agents plays the same role as an external field in traditional equilibrium phase transitions. We found that the criticality depends on the way groups of investors are formed according to different geometric arrangements. In regular lattices, only the dimension of the lattice and not the microscopic neighborhood details seems to define the critical behavior, in the same way observed in equilibrium transitions. In non-regular, fractal lattices, however, the scenario seems to be not so simple, again like equilibrium transitions. I am grateful to Hans Jürgen Herrmann for a critical reading of the manuscript. During the evaluation process, I was informed that my good friend and collaborator Dietrich Stauffer passed away. With consternation and very unhappy, I dedicate this work to him. References [1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971; J.J. Binney, N.J. Dowrick, A.J. Fisher, E.J.M. Newman, The Theory of Critical Phenomena, Oxford University Press, 1993. [2] P.M.C. de Oliveira, Europhys. Lett. 119 (2017) 40007. [3] As a curiosity, regressive taxation holds in almost all real economies, including all major countries. [4] In traditional thermodynamic transitions, a very small residual external field is necessary in order to break the symmetry in finite systems. Also here, due to the finite precision of real numbers in computers, a very small redistribution factor R is necessary, otherwise some agent could reach the limiting value wn = 0. After such an event, the corresponding agent would be forever excluded off the game. In practice, we replace wn by 10-300 every time it reaches this bottom limit. [5] V.M. Yakovenko, J.B. Rosser jr, Rev. Modern Phys. 81 (2009) 1703; B.K. Chakrabarti, A. Chakraborti, S. Chakravarty, A. Chatterjee, Econophysics of Income and Wealth Distributions, Cambridge University Press, 2013. [6] J.-P. Bouchaud, M. Mézard, Physica A 282 (2000) 536. [7] M. Levy, J. Econom. Theory 110 (2003) 42. [8] C.F. Mourkazel, J. Stat. Mech. (2011) P08023; C.F. Moukarzel, S. Gonçalves, J.R. Iglesias, M. Rodríguez-Achach, R. Huerta-Quintanilla, Eur. Phys. J. - Special Topics 143 (2007) 75; J.R. Iglesias, M.C. R.de Almeida, Eur. Phys. J. B 85 (2012) 85, and references therein. [9] M.E.J. Newman, R.M. Ziff, Phys. Rev. Lett. 85 (2000) 4104; P.M.C. de Oliveira, R.A. Nóbrega, D. Stauffer, Braz. J. Phys. 33 (2003) 616; J. Phys. A 37 (2004) 3743.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Investment/taxation model: Investors in groups P.M.C. de Oliveira
∗
Instituto de Física, Universidade Federal Fluminense, Niterói - RJ, 24210-340, Brazil Universidade Federal da Integração Latino Americana, Foz do Iguaçu - PR, 85867-000, Brazil National Institute of Science and Technology - Complex Systems, Brazil
article
info
Article history: Received 7 June 2019 Received in revised form 15 August 2019 Available online 16 September 2019
a b s t r a c t Individuals invest their capital and/or working skills, multiplying their wealths. This process is modeled by grouping investors according to different network possibilities. They also pay taxes according to progressive tax rates (increasing for increasing wealths) or regressive (the contrary). Independent of the particular rule of investment, there is a dynamically induced transition where all the population wealth falls in hands of a single individual for regressive taxation. In this case, the economic evolution is extinct. On the other hand, for progressive taxation the economic evolution continues forever. The critical properties of this transition, however, depend on the geometric arrangement adopted for investments. Here, the transition is studied in different spatial dimensions. For regular lattices a dimension universality is found, where the critical properties depend only on the lattice dimension. © 2019 Elsevier B.V. All rights reserved.
The main plot in Fig. 1, to be discussed later, shows what physicists call a critical quantity. Its importance, both theoretically as well as in numerous practical applications (a myriad of electronic devices for instance) resides in the possibility of controlling this quantity along the vertical axis by fine tuning its control parameter along the horizontal axis. Thanks to the tangency of the curve with the vertical axis, and the consequent mathematical singularity at the critical point. It is also the signature of continuous phase transitions (magnetic materials for instance), the behavior of the system abruptly changes when the critical point is crossed. Also, near the critical point, the behavior of these systems is governed by macroscopic concerns instead of details of its microscopic constituents, leading to the phenomenon of universality: completely different systems (for instance, an uniaxial ferromagnetic material near its magnetic transition temperature and a fluid like water near 374C, above which the coexistence of vapor and liquid phases no longer exists) present exactly the same critical exponent β for the respective quantities (the magnetization in the first case or the difference between liquid and vapor densities in the other). Due to this universality phenomenon, very simplified theoretical models sharing the same few macroscopic features with the complicated system under study can be applied with success in order to predict its macroscopic properties. For a review, see [1]. Although the theory of continuous phase transitions, critical phenomena and universality was well stablished for systems in thermodynamic equilibrium, its features were observed also in other out-of-equilibrium systems. Even in cases where thermodynamic concerns are absent, not only in the domain of Physics but also in many other domains like Biology, Ecology, Chemistry, Sociology, Economics, i.e. dynamic evolutionary systems in general. Nowadays, the strategy of applying very simplified dynamic models in order to study phenomena observed in these domains is widespread. In [2] the economic evolution of a society is modeled with two dynamic ingredients, repeated every year: (I) Individual agents invest their capital or work potential, altogether represented by their ‘‘wealths’’, Wn (n = 1, 2 . . . N); (II) They pay taxes. Step I is a simple multiplication of Wn by a random factor tossed from an arbitrary but fixed probability distribution. ∗ Correspondence to: Instituto de Física, Universidade Federal Fluminense, Niterói - RJ, 24210-340, Brazil. E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.122588 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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P.M.C. de Oliveira / Physica A 537 (2020) 122588
Fig. 1. Critical behavior: A tiny tuning of the control parameter p, near the critical point (here p = pc = 0), triggers a huge but continuous jump of the order parameter m. A phase transition occurs when the critical point is surpassed, from an inactive phase (m = 0 on the lefthand side) towards an active phase (m > 0 on the righthand side). The curve is tangent to the vertical axis at the critical point, its mathematical form is m ∼ pβ , with critical exponent β < 1 (here β = 0.64). The error bars are smaller than the symbols. Inset: The singularity at pc disappears when a uniform external field is applied (here, h = 0.1, 0.08, 0.04 and 0.025, smooth curves from top down).
In step II a tax rate depending linearly on the wealth Wn is adopted, defined by a fixed parameter p, the angular coefficient (slope) of the quoted linear dependence. For p > 0 the taxation is progressive, i.e. rich agents ∑ pay larger tax rates than poor agents. For p < 0 it is regressive, the contrary [3]. The wealth shares wn = Wn /S, where S = n Wn is the total population wealth, are considered. In particular, the largest share w1 corresponding to the richest agent. Also, these shares are always considered in decreasing order, so the index n does not correspond to one specific agent, but to its position along the current ranking of wealths. For each fixed p this dynamic rule is applied iteratively, year after year. For regressive taxation (p < 0), after some time the total population wealth is concentrated in the hands of a single agent, w1 = 1 with wn = 0 for all other agents n > 1. In this regressive case, the economic evolution of the society stops, it is what physicists call an absorbing state of the dynamics, biologists would say extinction. For progressive taxation (p > 0), the distribution of wealth shares fluctuates forever, but after some time their time averages over the last years reaches a steady state with wn < 1 for all agents. Therefore, there is a dynamically induced transition from activity towards inactivity (absorbing state) when one crosses the frontier p = pc = 0 from progressive to regressive taxation. The order parameter describing this transition is m = −⟨log w1 ⟩ where ⟨. . . ⟩ represents the time average in the steady state, as well as a further average over different realizations. The transition occurs always at the frontier p = pc = 0 separating progressive from regressive taxation, for numerous different rules of investment, step I. Fig. 1 shows the order parameter m as a function of the control parameter p for one such rules. In short, the model consists of the following steps, repeated each year. ∑ Step I (investment). The wealth shares w1 > w2 > w3 > · · · > wN are normalized, n wn = 1. Each one is multiplied by a random factor fn obtained from an arbitrary but fixed rule, Wn = fn wn
∑
and the new sum S = n Wn is calculated. Step II (taxes). Each agent pays taxes according to a linear tax rate Wn′ = Wn (1 − A − p Wn /S) where the extra parameter A plays no role, it is included only to assure the paid tax is positive and not larger ∑than the wealth, 0 < A < 1 and 0 < A + p < 1. Before starting the new year, the current wealth shares wn′ = Wn′ / i Wi′ are calculated in order to be used in the next year. This renormalization of wealths can be interpreted as a simple redefinition of the monetary unit (currency), always taken as the total population wealth at the beginning of each year.
P.M.C. de Oliveira / Physica A 537 (2020) 122588
3
The order parameter m = −⟨log w1 ⟩ has an interesting interpretation within the Economics point of view, related to social inequalities. The larger it is the smaller are these inequalities, i.e. the smaller is the ratio between the richest agent and the average wealth share 1/N. Within this interpretation, even a small degree of progressive taxation through a small positive value for the control parameter p is effective in mitigating inequalities, thanks to the critical behavior shown in Fig. 1 and its characteristic tangency between the curve and the vertical axis. Also, the smaller the critical exponent β < 1 the larger is this effectiveness. Besides progressive taxation, another traditional way to mitigate social inequalities is a redistribution of the paid taxes to all individuals. An extra step III can be introduced into the above rules, before the wealth distribution renormalization. Step III (redistribution). A fraction R of the total taxes
∑
Wn (A + p Wn /S)
n
paid by the population is uniformly redistributed among all agents. This ingredient plays the same role of a uniform external field applied to traditional thermodynamic systems, namely to turn critical, singular curves like the main plot in Fig. 1 into smooth curves as shown in the inset of the same Fig. 1. Here, the external field is measured by a further control parameter h defined below 1 h
= − log R
representing the degree of redistribution. The critical situation corresponds to h → 0, no redistribution. Following this trend, the smooth curves tend to the critical one [4]. This model inserts into a class of similar others, references [5] give good reviews and many further references. In particular, the pioneering work [6] worths to be a basis for comparison. The investment step I, a multiplicative random process, is the same. However, a particular probability (log normal) distribution for the factors fn was adopted in this work, whereas the current model relaxes this restriction. Indeed, any probability distribution for the investment factors fn produces a Pareto distribution of wealths among the population, provided it is the same for all agents. This point is formally demonstrated in [7], in agreement with the observed behavior of numerous real markets. Here, a similar universality is observed concerning the phase transition, always occurring at p = pc = 0. Also, the taxation step II takes into account only a uniform taxation rule in [6], i.e. only the parameter A is present (p = 0), therefore the transition could not be captured. Also the criticality could not be captured, since only the full redistribution (R = 1 or h → ∞) is taken into account. We tested also the inclusion of a pair-coupling Jij where agents i and j trade among themselves, like [6]. One such agent (payer) transfers a fraction of its wealth to the other (receiver). The overall result of this coupling is thus to transfer wealth from the richest agents to the poorest agents. The effect of this coupling is thus similar to the inclusion of the already quoted external field, collected taxes uniformly redistributed. Therefore, it is not considered in the numerical results shown here. The spatial correlations among agents induced by some network is present only in the investment grouping mechanism explained later. Concerning the control of social inequalities, within similar models, some references are also provided [8]. Among the numerous investment rules one can invent, every one leading to the dynamically induced transition at pc = 0, a simple one is to take half the agents, doubling their wealths (fn = 2) while the other half is untouched (fn = 1). One can interpret this as a random walk in the would-be ‘‘space’’ of N agents repeated until covering a number N /2 of them (each agent covered by the random walk has its wealth doubled only once, even when revisited). In [2] this half-half division would be taken at random, which corresponds to a random walk over an infinite-dimensional hyper-cubic lattice, a network of N sites every one linked to any other. The purpose of the current work is to study other possible networks, to investigate how the links forming groups of investors acting together would be reflected into the transition criticality. A first possibility is to consider the agents positioned along a 1-dimensional ring with N sites. This rule corresponds to Fig. 1. The same rule generates the steepest straight line in Fig. 2, no longer tangent to the vertical axis. It means that progressive taxation (Fig. 1) is more effective than redistribution (Fig. 2) when the purpose is to mitigate inequalities. An alternative is to place the agents on a 2-dimensional L × L square lattice (N = L2 ), tossing N /2 agents through a random walk starting from a random site, and doubling their wealths. The equivalent to Fig. 1 (not shown) is similar with a larger critical exponent. Another alternative is a triangular lattice (instead of square), also 2-dimensional. An interesting observation is the coincidence (within the error bars) of results on the square or triangular lattices. In particular, both share the same exponent β . Also as function of the external field, the middle straight line on Fig. 2 exhibits this coincidence. Again by analogy with traditional thermodynamic phase transitions, they seem to belong to the same ‘‘universality class’’ which would be determined only by the geometrical dimension of the network where the agents are located. This behavior implies that somehow to approach the critical situation p = h = 0 corresponds to increase the geometrical length scale relevant to the observed criticality. Following this trend, this length scale becomes much larger than the microscopic lattice details. In order to confirm this behavior, Fig. 3 shows the results for different 3-dimensional lattices. Indeed, the coincidence occurs again when the critical point (p = h = 0) is approached from both directions. A larger exponent β is shared by all 3-dimensional cases. In infinite dimension (the random case) the exponent is larger yet. In all these cases of regular lattices, plots like Fig. 2 result in asymptotically straight lines towards the origin, i.e. the mathematical form is m = ah1 , with decreasing slopes. Numerical results are displayed in Table 1.
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P.M.C. de Oliveira / Physica A 537 (2020) 122588
Fig. 2. Order parameter as a function of the external field at the critical point p = pc = 0. Top: The same one-dimensional ring as in Fig. 1. Middle: 2-dimensional square and triangular lattices (different colors on line). Bottom: Fractal cluster constructed over a square lattice by erasing random sites, at the percolation threshold concentration 0.59274621 of remainder sites.
Fig. 3. Order parameter as a function of the external field at the critical point p = pc = 0, for 3-dimensional lattices. Simple cubic (open circles, straight line), body-centered cubic (filled circles) and face-centered cubic (squares).
Worth to mention that even in the absence of the coupling Jij , like [6], the lattice geometry plays a fundamental role because the set of neighbor agents investing together, i.e. the whole set under the same investment random factor (step I), is defined within this geometry by the random walk. Regular lattices are obviously not realistic networks for groups of investors (clients of the same investment firm for stock market, or members of the same workers representative institution, for example). More realistic networks presenting fractal properties could be tested. However, this is not the purpose of the current work, but instead to explore the similarities of this non-equilibrium transition with the traditional behavior of thermodynamic equilibrium transitions, in particular the phenomenon of lattice dimension universality. Anyway, an example of non-regular, fractal network is
P.M.C. de Oliveira / Physica A 537 (2020) 122588
5
Table 1 Critical exponent β of the order parameter m ∼ pβ without redistribution (h → 0) as in the main plot of Fig. 1, and slope a of the straight line m = ah at the critical point (p = pc = 0) as in Fig. 2, in 1, 2, 3 and ∞ dimensions. Error bars are smaller than 10−2 in all cases. 1D
2D
3D
∞D
β = 0.64 a = 7.20
0.80 1.35
0.84 1.18
0.85 1.16
tested: percolation clusters obtained by diluting the sites of a square lattice at the percolation threshold, i.e. a fraction 0.59274621 of present sites [9]. One example is shown in Fig. 2, bottom curve. In this case one can observe the tangency occurring between the curve and the horizontal axis (instead of vertical as in Fig. 1), meaning that the strategy of redistribution is very inefficient when the purpose is to mitigate inequalities. The mathematical form is m ∼ he with the exponent e > 1 (e ≈ 1.3 in Fig. 2, bottom curve). As a function of p, however, an intermediate behavior between that of 1 and 2 dimensions was found. Concluding, an investment/taxation model was introduced with investors acting together in groups. Each year, after the investment step, agents pay taxes according to a progressive or regressive taxation. This iterative dynamic process leads to a phase transition: for regressive taxation the dynamic evolution stops with the entire population wealth in hands of a single agent, whereas for progressive taxation the economy evolves forever with the population wealth distributed among all agents. The wealth share w1 corresponding to the richest agent, averaged in time and realizations, defines the order parameter m = −⟨log w1 ⟩ for this transition, with m = 0 for regressive taxation. For progressive taxation, corresponding to a positive control parameter p, the critical behavior of the transition can be determined. A second control parameter h related to a uniform redistribution of paid taxes among all agents plays the same role as an external field in traditional equilibrium phase transitions. We found that the criticality depends on the way groups of investors are formed according to different geometric arrangements. In regular lattices, only the dimension of the lattice and not the microscopic neighborhood details seems to define the critical behavior, in the same way observed in equilibrium transitions. In non-regular, fractal lattices, however, the scenario seems to be not so simple, again like equilibrium transitions. I am grateful to Hans Jürgen Herrmann for a critical reading of the manuscript. During the evaluation process, I was informed that my good friend and collaborator Dietrich Stauffer passed away. With consternation and very unhappy, I dedicate this work to him. References [1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971; J.J. Binney, N.J. Dowrick, A.J. Fisher, E.J.M. Newman, The Theory of Critical Phenomena, Oxford University Press, 1993. [2] P.M.C. de Oliveira, Europhys. Lett. 119 (2017) 40007. [3] As a curiosity, regressive taxation holds in almost all real economies, including all major countries. [4] In traditional thermodynamic transitions, a very small residual external field is necessary in order to break the symmetry in finite systems. Also here, due to the finite precision of real numbers in computers, a very small redistribution factor R is necessary, otherwise some agent could reach the limiting value wn = 0. After such an event, the corresponding agent would be forever excluded off the game. In practice, we replace wn by 10-300 every time it reaches this bottom limit. [5] V.M. Yakovenko, J.B. Rosser jr, Rev. Modern Phys. 81 (2009) 1703; B.K. Chakrabarti, A. Chakraborti, S. Chakravarty, A. Chatterjee, Econophysics of Income and Wealth Distributions, Cambridge University Press, 2013. [6] J.-P. Bouchaud, M. Mézard, Physica A 282 (2000) 536. [7] M. Levy, J. Econom. Theory 110 (2003) 42. [8] C.F. Mourkazel, J. Stat. Mech. (2011) P08023; C.F. Moukarzel, S. Gonçalves, J.R. Iglesias, M. Rodríguez-Achach, R. Huerta-Quintanilla, Eur. Phys. J. - Special Topics 143 (2007) 75; J.R. Iglesias, M.C. R.de Almeida, Eur. Phys. J. B 85 (2012) 85, and references therein. [9] M.E.J. Newman, R.M. Ziff, Phys. Rev. Lett. 85 (2000) 4104; P.M.C. de Oliveira, R.A. Nóbrega, D. Stauffer, Braz. J. Phys. 33 (2003) 616; J. Phys. A 37 (2004) 3743.