# An *H* Theorem for Boltzmann’s Equation for the Yard-Sale Model of Asset Exchange

*H*Functional

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## Abstract

In recent work (Boghosian, Phys Rev E 89:042804–042825, 2014; Boghosian, Int J Mod Phys 25:1441008–1441015, 2014), Boltzmann and Fokker–Planck equations were derived for the “Yard-Sale Model” of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an *H* function of both the Boltzmann and Fokker–Planck equations for the model.

## Keywords

Boltzmann equation Asset exchange model Yard-Sale model*H*theorem Gini coefficient Pareto distribution

## 1 Introduction

Over 100 years ago, the Italian economist Vilfredo Pareto [3] made one of the first empirical studies of the distribution of wealth by undertaking a careful study of land ownership in Italy, Switzerland and Germany. In the course of this study, he plotted the fraction of economic agents^{1} with land holdings worth more than *w* as a function of *w*. His studies led him to believe that this function, which we shall denote by *A*(*w*) has a universal form.

*A*is monotone non-increasing, and that \(A(0)=1\) and \(\lim _{w\rightarrow \infty }A(w)=0\). What Pareto observed is that

*A*(

*w*) is approximately equal to one for all

*w*less than a certain cutoff value denoted by \(w_{\min }\), and decays as a power law for \(w > w_{\min }\). That is, Pareto’s empirical observations led him to conclude that it is approximately true that

To put Pareto’s observations in modern terms, we may note that \(1-A(w)\) is the cumulative distribution function (CDF) of economic agents, ordered by wealth. Let us denote the corresponding probability density function (PDF) of agents by *P*(*w*), but we shall adopt the convention of normalizing *P* to the total number of economic agents, rather than to unity, so that \(\int _a^b dw\; P(w)\) is the total number of agents with wealth in [*a*, *b*], for \(0\le a<b\).

^{2}

## 2 The Lorenz Curve

*L*(

*w*) and

*F*(

*w*) are monotone nondecreasing, and they increase from zero to one as

*w*goes from zero to infinity. This graph, which has come to be called the

*Lorenz curve*, can be plotted in the unit square, as shown schematically in Fig. 1.

If all wealth in a society were equally distributed, the Lorenz curve would be the diagonal line, plotted in blue in Fig. 1. This is because any fraction \(f\in [0,1]\) of the agents would possess the same fraction *f* of the total wealth. In real economies, this curve always lies below the diagonal, more like the orange curve in Fig. 1.

*w*corresponding to that point on the curve to the average wealth. It follows that

*L*is now considered a function of

*F*. We see that \(L=F=0\) when \(w=0\), and that \(L=F=1\) when \(w\rightarrow \infty \), and moreover that

*L*(

*F*) is concave up. It follows that

*L*(

*F*) is bounded above by the diagonal.

*P*(

*w*), and we may also note that

*N*and

*W*, we see that any of the quantities

*A*(

*w*),

*P*(

*w*),

*L*(

*w*) and

*F*(

*w*) can be derived from any other, so they all contain equivalent information – which is to say that they all contain essentially complete information about the distribution of wealth in a society.

## 3 The Gini Coefficient

In fact, actual data for wealth distributions in the world today is very scant, and economists have to content themselves with much coarser characterizations of wealth inequality than the quantities described above. One of the most popular of these is due to the Italian statistician and sociologist Corrado Gini, who also worked roughly contemporaneously with Pareto and Lorenz.

The *Gini coefficient* *G* is defined as the ratio of the shaded area in Fig. 1, lying between the diagonal segment and the Lorenz curve. Consequently, \(G=0\) when everybody has equal wealth; the limit \(G\rightarrow 1\) describes the approach to complete oligarchy.

*F*to

*w*to find

*w*in the rightmost expression. Equation (6) indicates that \(G=1-2\langle L\rangle \), where the angle brackets denote an average over the PDF

*P*. Likewise, Eq. (7) indicates that \(G=1-2(N/W)\langle Aw\rangle \). From these fundamental relationships between the Gini coefficient and the functions introduced earlier, we see that

*G*is a quadratic functional of

*P*, and we may sometimes emphasize this functional dependence by writing it as

*G*[

*P*].

*Fréchet derivative*of

*G*[

*P*] with respect to

*P*, which is the analog of the gradient in function space.

^{3}This is defined by noting that, for any sufficiently well behaved function \(\eta (w)\),

*g*.

*P*changes in time, it will cause a change in the Gini coefficient given by

*P*, and use them along with Eq. (9) to place bounds on

*dG*/

*dt*.

## 4 Background to the Yard-Sale Model of Asset Exchange

As noted in the introduction, though Pareto’s law is over a century old, an explanation for it in terms of microeconomic exchange relations between individual agents is still elusive. The general idea that simple rules for asset exchange might be used to explain wealth distributions appears to be due to Angle [5] in 1986. Such models have come to be called *Asset-Exchange Models* (AEMs) and they typically involve binary transactions between agents involving some increment of wealth \(\varDelta w\), with rules for which agent gains it and which agent loses it. The first work applying the mathematical methods of kinetic theory to such models appears to be the paper of Ispolatov, Krapivsky and Redner [6] in 1998. They considered an AEM model in which the agent who loses the wealth is selected with even odds, and in which \(\varDelta w\) is proportional to the wealth of the losing agent. Writing in a popular article in 2002, Hayes [7] noted that in an economy governed by this model, no agent would willingly trade with a poorer agent, and therefore would try to use deception to trade only with wealthier agents. For this reason, he named the model of Ispolatov et al. the *Theft-and-Fraud Model* (TFM).

In 2002, Chakraborti [8] introduced a variant of the TFM, in which the losing agent is still selected with even odds, but in which \(\varDelta w\) is proportional to the wealth of the poorer agent, rather than that of the losing agent. In the same popular article mentioned above, Hayes [7] referred to this as the *Yard-Sale Model* (YSM), and noted that it describes an economy in which rich and poor will have no strategic reason not to trade with each other. The first kinetic theoretical description of the YSM was given by Boghosian in 2014 [1, 2]. In analogy with [6], he derived a Boltzmann equation for the YSM, but then went on to show that in the limit of small \(\varDelta w\) and frequent transactions, this reduced to a certain nonlinear, nonlocal Fokker–Planck equation. He presented numerical evidence indicating that the YSM by itself exhibits “wealth condensation,” in which all the wealth ends up in the hands of a single oligarch. He also showed that, when supplemented with a simple model for redistribution, the YSM yields Pareto-like wealth distributions, including a cutoff at low wealth and an approximate power law at large wealth, very reminiscent of Eq. (1).

The YSM can be described by a very simple algorithm. The version of the algorithm we shall use here is completely equivalent to that described by Boghosian [1, 2], though we state it in the following slightly different fashion: Choose two agents from the population at random to engage in a financial transaction. Call them agent 1 and agent 2, and denote their respective wealth values by \(w_1\) and \(w_2\). The amount of wealth that will be transferred from agent 1 to agent 2 in this transaction is then \(\varDelta w = \beta \,\text{ min }(w_1,w_2)\), where \(\beta \in (-1,+1)\) is sampled from a *symmetric* PDF denoted by \(\eta (\beta )\). Note that, because \(\eta (\beta )\) is symmetric, the two agents both have even odds of winning and losing.

It may seem strange that an algorithm which gives even odds of winning to both agents engaging in a transaction would cause wealth to concentrate, but this is indeed the case. This was demonstrated by extensive numerical simulations in [1], and there it was conjectured that the time-asymptotic state of *P*(*w*) is a generalized function, with support at \(w=0\), zeroth and first moments given by *N* and *W* respectively, and possibly divergent higher moments. This corresponds to an oligarchical state with \(G=1\). In this paper, we confirm that conjecture by demonstrating that the Gini coefficient is a monotone increasing Lyapunov functional of the Boltzmann equation for the YSM, and that it reaches a maximum value of \(G=1\) in the above-described oligarchical state.

## 5 The Boltzmann Equation of the Yard-Sale Model

*G*is monotone non-decreasing as a consequence of the above dynamics for

*P*. The Lyapunov function of the molecular Boltzmann equation is traditionally called Boltzmann’s “

*H*function.” Adopting that nomenclature, we will show that the Gini coefficient is an

*H*function for the Yard-Sale Model Boltzmann equation.

*dG*/

*dt*thereby obtained is greater than or equal to zero. We partition this task by rewriting the above as

*u*back to

*w*to obtain

*w*, as follows from

*y*is outermost, and then make the substitution \(u=w-\beta y\). In the second and fourth terms, use the change of variable \(u=w/(1+\beta )\), and then swap the order of integration so that

*y*is outermost. The result is

*x*is outermost, followed by

*y*, and then

*u*to obtain

*u*from

*y*to \(x-\beta y\) can be split into one from

*y*to \(x/(1+\beta )\) plus another from \(x/(1+\beta )\) to \(x-\beta y\), resulting in

## 6 Fokker–Planck Equation for the Yard-Sale Model

*G*is also a Lyapunov functional of this equation. Here we demonstrate this directly.

## 7 The Time-Asymptotic State of the Yard-Sale Model

It is well known that the equilibrium solution of the molecular Boltzmann equation, namely the Maxwell-Boltzmann distribution, may be found by setting the variation of Boltzmann’s *H* function to zero, under the constraints of fixed mass, momentum and energy. We may attempt an analogous computation here, but, as conjectured by Boghosian [1], the equilibrium solution of the Boltzmann equation for the Yard-Sale Model without redistribution is a singular generalized function, \(\zeta (w)\). It is zero for \(w\ne 0\), has zeroth moment *N* and first moment *W*, and its higher moments may not exist. It is only when redistribution is included that steady-state solutions similar to the Pareto distribution, Eq. (1), are obtained [1]. Still, it is instructive to see if the variational approach will yield this singular generalized function, so we turn our attention to that problem in this last section.

*G*[

*P*], we introduce the Lagrange multipliers \(\lambda \) and \(\mu \) to enforce the constraints

*P*. Using Eqs. (5), (6) and (7), we obtain

*P*, and using Eq. (7), we obtain

*G*increases in time and asymptotically approaches the value one, corresponding to a state of “perfect oligarchy.”

## 8 Conclusions

We have proven that the Gini coefficient *G* is a Lyapunov function of the Boltzmann equation for the Yard-Sale Model of asset exchange, as well as of the Fokker–Planck equation obtained in the limit of small transaction sizes. We have also shown that the equilibrium distribution, obtained in the time-asymptotic limit, is zero for all \(w\ne 0\), and corresponds to \(G=1\).

As noted earlier, it is only when the Yard-Sale Model is supplemented with a mechanism for redistribution that steady states similar to the Pareto distribution are found. With redistribution, however, the Gini coefficient is clearly no longer a Lyapunov functional, since it would be possible to begin with a higher concentration of wealth than that obtained in the time-asymptotic limit. It may be possible to find a different Lyapunov functional for the Boltzmann or Fokker–Planck equations with a redistribution term, but we leave this as a topic for future study.

## Footnotes

- 1.
Throughout this paper, we use the term “economic agent” to refer to any entity that can hold or exchange wealth. This includes, but is not limited to individuals, corporations, funds, central banks, etc. By “wealth,” we refer to currency or anything that can be bought or sold with currency. This includes, but is not limited to, real estate, stock, commodities, factories, etc.

- 2.
Here and throughout, we assume that derivatives of these quantities with respect to

*w*exist, at least in the distributional sense. - 3.
See e.g., [4].

## References

- 1.Boghosian, B.M.: Kinetics of wealth and the Pareto law. Phys. Rev. E
**89**, 042804–042825 (2014)CrossRefADSGoogle Scholar - 2.Boghosian, B.M.: Fokker–Planck description of wealth dynamics and the origin of Pareto’s law. Int. J. Mod. Phys. C
**25**, 1441008–1441015 (2014)CrossRefADSGoogle Scholar - 3.Pareto, V.: The Mind and Society—Trattato Di Sociologia Generale. Harcourt, Brace (1935)Google Scholar
- 4.Frigyik, B.A., Srivastava, S., Gupta, M.R.: An introduction to functional derivatives. UWEE Technical Report No. UWEETR-2008-0001 (2008)Google Scholar
- 5.Angle, J.: The surplus theory of social stratification and the size distribution of personal wealth. Soc. Forces
**65**, 293–326 (1986)CrossRefGoogle Scholar - 6.Ispolatov, S., Krapivsky, P.L., Redner, S.: Wealth distributions in asset exchange models. Eur. Phys. J. B Condens. Matter
**2**, 267–276 (1998)Google Scholar - 7.Hayes, B.: Follow the money. Am. Sci.
**90**, 400–405 (2002)CrossRefGoogle Scholar - 8.Chakraborti, A.: Distributions of money in model markets of economy. Int. J. Mod. Phys. C
**13**, 13151321 (2002)Google Scholar

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