Wealth distribution: To be or not to be a Gamma?

Wealth distribution: To be or not to be a Gamma?

Mehdi Lallouache mehdi.lallouache@ens-cachan.fr Département de Physique, École Normale Supérieure de Cachan, 94230 Cachan, France Chaire de Finance Quantitative, Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, 92290 Châtenay-Malabry, France    Aymen Jedidi aymen.jedidi@ecp.fr Chaire de Finance Quantitative, Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, 92290 Châtenay-Malabry, France    Anirban Chakraborti anirban.chakraborti@ecp.fr Chaire de Finance Quantitative, Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, 92290 Châtenay-Malabry, France
July 13, 2019
Abstract

We review some aspects, especially those we can tackle analytically, of a minimal model of closed economy analogous to the kinetic theory model of ideal gases where the agents exchange wealth amongst themselves such that the total wealth is conserved, and each individual agent saves a fraction () of wealth before transaction. We are interested in the special case where the fraction is constant for all the agents (global saving propensity) in the closed system. We show by moment calculations that the resulting wealth distribution cannot be the Gamma distribution that was conjectured in Phys. Rev. E 70, 016104 (2004). We also derive a form for the distribution at low wealth, which is a new result.

Econophysics; Gamma distribution; kinetic theory
pacs:
89.65.Gh 87.23.Ge 02.50.-r

I Introduction

The distribution of wealth or income in society has been of great interest for many years. As first noticed by Pareto in the 1890’s Pareto (), the wealth distribution seems to follow a “natural law” where the tail of the distribution is described by a power-law . Away from the tail, the distribution is better described by a Gamma or Log-normal distribution known as Gibrat’s law Gibrat (). Considerable investigation with real data during the last ten years revealed that the power-law tail exhibits a remarkable spatial and temporal stability and the Pareto index is found to have a value between and Dragulescu2001 (); Aoyama2003 (). Even after 110 years the origin of the power-law tail remained unexplained but recent interest of physicists and mathematicians in econophysics has led to a new insight into this problem (see Refs. Yakovenko2009 (); Arnab2007 (); Chakrabarti2010 ()).
Our general aim is to study a many-agent statistical model of closed economy (analogous to the kinetic theory model of ideal gases) Bennati (); Angle (); Ispolatov1998 (); Dragulescu2000a (); Chakraborti2000a (); Chatterjee2003a (), where agents exchange a quantity , that may be defined as wealth. The states of agents are characterized by the wealth , and the total wealth is conserved. The evolution of the system is then carried out according to a prescription, which defines the trading rule between agents. These many-agent statistical models have basic units , interacting with each other through a pair-wise interaction characterized by a saving parameter , with . We define the equilibrium distribution of wealth as follows : is the probability that in the steady state of the system, a randomly chosen agent will be found to have wealth between and . In these models, if is equal for all the units, is fitted quite well by a Gamma-distribution Anirban2008 (); Patriarca2004a (); Patriarca2004b ()

(1)

where

(2)

This equilibrium distribution (1) had been suggested by an analogy with the kinetic theory of gases in dimensions Anirban2008 (); Patriarca2004a (); Patriarca2004b ().

In this paper we show by the method of moment calculations that the resulting wealth distribution cannot be the Gamma distribution that was conjectured in Ref. Patriarca2004a (); Patriarca2004b (). We also derive the functional form of an upper bound on at very small .

Ii Many-agent model of a closed economy

We study many-agent statistical models of closed economy (analogous to the kinetic theory model of ideal gases), where agents exchange wealth . The states of agents are characterized by the wealth , and the total wealth is conserved. The evolution of the system is then carried out according to a prescription, which defines the trading rule between agents. At every time step two agents and are extracted randomly and an amount of wealth is exchanged between them,

(3)

It can be noticed that in this way, the quantity is conserved during the single transactions: (see Fig. 1), where and are the agent wealths after the transaction has taken place. Several simple models dealing with different transaction rules have been studied (see the reviews Yakovenko2009 (); Arnab2007 (); Patriarca2010 () and references therein). Here we will present a few examples.

ii.1 Basic model without saving: Boltzmann distribution

In the first version of the model, the wealth difference is assumed to have a constant value Bennati (),

(4)

This rule, together with the constraint that transactions can take place only if and , provides a Boltzmann distribution, see the curve for in Fig. 2. Alternatively, can be a random fraction of the wealth of one of the two agents,

(5)

where is a random number uniformly distributed between 0 and 1. A trading rule based on the random redistribution of the sum of the wealths of the two agents had been introduced by Dragulescu and Yakovenko Dragulescu2000a (),

(6)

Equations (6) are easily shown to correspond to the trading rule (3), with

(7)

All the versions of the basic model lead to an equilibrium Boltzmann distribution, given by

(8)

where the effective temperature of the system is just the average wealth Bennati (); Dragulescu2000a (). The result (8) is found to be robust; it is largely independent of various factors. Namely, it is obtained for the various forms of mentioned above, for a pair-wise as well as multi-agent interactions, for arbitrary initial conditions Chakraborti2000a (), and finally, for random or consecutive extraction of the interacting agents. For the trading rule (6) one can show the convergence towards the Boltzmann distribution through different methods: Boltzmann equation, entropy maximization, distributional equation, etc.

Figure 1: Analogy of the minimal economic model with a classical isolated system of ideal gas, where the particles are randomly undergoing “Elastic” collisions, and exchanging kinetic energy. In the closed economy, the economic agents randomly trade with each other according to some rule and exchange wealth.
Figure 2: Probability density for wealth . The curve for is the Boltzmann function for the basic model of Sec. II.1. The other curves correspond to a global saving propensity (see Sec. II.2).

ii.2 Model with global saving propensity

A step toward generalizing the basic model and making it more realistic, is the introduction of a saving criterion regulating the trading dynamics. This can be practically achieved by defining a saving propensity , which represents the fraction of wealth which is saved – and not reshuffled – during a transaction. The dynamics of the model is as follows Chakraborti2000a (); Chakraborti2002a ():

(9)

corresponding to a in Eq. (3) given by

(10)

This model leads to a qualitatively different equilibrium distribution. In particular, it has a mode and a zero limit for small , see Fig. 2. Later we will derive a form for an upper bound on at low range. The functional form of such a distribution was conjectured to be a -distribution, as given by Eq. (1) on the basis of an analogy with the kinetic theory of gases. Indeed, it is easy to show, starting from the Maxwell-Boltzmann distribution for the particle velocity in a dimensional gas, that the equilibrium kinetic energy distribution coincides with the Gamma-distribution (1) with . This conjecture is remarkably consistent with the fitting provided to numerical data Anirban2008 (); Patriarca2004a (); Patriarca2004b (). In the following section we will show by two different approaches that the conjecture (1) cannot be the actual equilibrium distribution.

Iii Analytical results for model with saving propensity

iii.1 Fixed-point distribution

Let be a random variable which stands for the wealth of one agent, at equilibrium and in the limit of an infinite number of agents, we can say from Eq. (9) that the law of , is a fixed-point distribution of the equation

(11)

where means identity in distribution and one assumes that the random variables and have the same probability law, while the variables and are stochastically independent. It seems difficult to find the distribution of , however, one can compute the moments of . Indeed with (11), one can write immediately

(12)

and by developing (12) one can find the recursive relation

(13)

Using (13) with initial conditions (normalization) and (without loss of generality), we obtain

(14)
(15)
(16)

Now let us compare theses moments with conjecture (1)’s moments. Setting in Eq. (1) it is easy to show

(17)

Writing (17) for and choosing as in (2) we find

(18)
(19)
(20)

The fourths moments (eqs.(16) and (20)) are different so the conjecture that the Gamma distribution is an equilibrium solution of this model is wrong. Nevertheless the first three moments coincide exactly which shows that the Gamma-distribution is strangely a very good approximation. Moreover the deviation in the fourth moment is very small (see Fig. 3, which shows that the two curves can hardly be distinguished by the naked eye).

Figure 3: Exact fourth moment eq.(16) and Gamma distribution fourth moment eq.(20) against . The inset shows the relative difference between exact fourth moment eq.(16) and Gamma distribution fourth moment eq.(20) against .

Finding a function that would coincide to higher moments is still an open challenge. These results are consistent with the ones found by Repetowicz et al. Repetowicz2005 () which will be presented in the following section.

iii.2 Laplace transform analysis

In this section we will confirm the previous result with a different approach based on the Boltzmann equation and along the lines of Bassetti et al. Matthes2008 (). Given a fixed number of agents in a system, which are allowed to trade, the interaction rules describe a stochastic process of the vector variable in discrete time . Processes of this type are thoroughly studied e.g. in the context of kinetic theory of ideal gases. Indeed, if the variables are interpreted as energies corresponding to the -th particle, one can map the process to the mean-field limit of the Maxwell model of particles undergoing random elastic collisions. The full information about the process in time is contained in the -particle joint probability distribution . However, one can write a kinetic equation for one-marginal distribution function

involving only one- and two-particle distribution functions

(21)

which may be continued to give eventually an infinite hierarchy of equations of BBGKY (Born, Bogoliubov, Green, Kirkwood, Yvon) type Plischke (). The standard approximation, which neglects the correlations between the wealth of the agents induced by the trade gives the factorization

which implies a closure of the hierarchy at the lowest level. In fact, this approximation becomes exact in the thermodynamic limit (). Therefore, the one-particle distribution function bears all information. Rescaling the time as in the thermodynamic limit , one obtains for the one-particle distribution function the Boltzmann-type kinetic equation

(22)

This equation can be written (see Matthes et al. Matthes2008 ()) as

where is a collision operator. A collision operator is bilinear and satisfies, for all smooth functions

(23)

where and are the post-trade wealth. With this property the equation can be written in the weak form, for all smooth functions

(24)

It is very useful because the choice gives (after some calculations) the Boltzmann equation for the Laplace transform of

(25)

For the steady state, and if is drawn randomly from a uniform distribution, the previous equation reduces to

(26)

which coincides with results of Repetowicz2005 (). The Taylor expansion of can be derived by substituting the expansion in (26). Since is the moment-generating function we have . With this method Repetowicz et al. Repetowicz2005 () obtained the recursive formula

(27)

with

(28)

Now with this formula one can obtain the first four moments and they match the ones found in the previous section eqs.(14-16), which confirms that the Gamma-distribution is not the stationary distribution.

Iv Upper bound form at low wealth range

From equation (11)

we have for all

(29)

where means the probability of the event inside the brackets. If the number of agents in the market is large, the distributions of different agents are independent. Then

(30)

where is the Heaviside step function. Taking the derivative with respect to in both sides, we have

(31)

This equation is an integral equation for . As mentioned earlier, we are not able to solve it in closed form. However, one can simplify the equation, by doing the integral over . Then the -function will contribute only if we have the following constraints

(32)
(33)
(34)

The range defined by these constraints is shown in figure 4. In this range, the derivative of the argument of the delta function with respect to is just . And, hence we get

(35)
Figure 4: Region of integration

This immediately gives

(36)

where

(37)

We assume that decays fast enough near 0 so that the integral in (37) is well defined. Now (36) may be rewritten by rescaling the variable, as

(38)

We now use the observation that for the numerically determined is a continuous function with a single maximum, say at (see Fig. 2). Then for all , the integrand (38) takes its maximum value at the right extreme point, i.e. when . This then gives us

(39)

Iterating this equation, we get

(40)

We can set in the above equation, giving

(41)

Then taking and rescaling the variables, we get

(42)

as , where and are two constants dependent on . The Gamma-distribution decays slower than the rhs in (42) when . The expression (42) gives an upper bound form at low wealth range and confirms again that the distribution of the global saving propensity model is not a Gamma-distribution.

V Discussion and outlook

We have used different approaches to show that the correct form of the wealth distribution cannot be the Gamma distribution. We have also derived an analytical form of an upper bound at low wealth range see Eq. (42). This is an analytically calculated upper bound but the closed form of the solution to Eq. (9) is still an open question.

As a further generalization, the agents could be assigned different saving propensities Chatterjee2003a (); Chatterjee2004a (); Chatterjee2005a (); Repetowicz2005 (); Patriarca2005a (); Chakraborti2009 (). In particular, uniformly distributed in the interval have been studied numerically in Refs. Chatterjee2003a (); Chatterjee2004a (). This model is described by the trading rule

(43)

or, equivalently, by a (as defined in Eq. (3)) given by

(44)

One of the main features of this model, which is supported by theoretical considerations Mohanty (); Chatterjee2005a (); Repetowicz2005 (), is that the wealth distribution exhibits a robust power-law at large values of ,

(45)

with a Pareto exponent largely independent of the details of the -distribution.

Acknowledgements.
The authors are grateful to D. Dhar for critical reading of the manuscript and inputs in finding the functional form of the wealth distribution at low range. The authors also acknowledge F. Abergel, N. Millot, M. Patriarca, M. Politi and A. Chatterjee for critical discussions or comments, and B.K. Chakrabarti for pointing out reference Chakrabarti2010 (). AC is grateful to Department of Theoretical Physics, TIFR for the kind hospitality where part of the work was initiated.

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