The non-universality of wealth distribution tails near wealth condensation criticality
TTHE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILSNEAR WEALTH CONDENSATION CRITICALITY
SAM L. POLK ∗ AND
BRUCE M. BOGHOSIAN † Abstract.
In this work, we modify the Affine Wealth Model of wealth distributions to exam-ine the effects of nonconstant redistribution on the very wealthy. Previous studies of this model,restricted to flat redistribution schemes, have demonstrated the presence of a phase transition to apartially wealth-condensed state, or “partial oligarchy”, at the critical value of an order parameter.These studies have also indicated the presence of an exponential tail in wealth distribution preciselyat criticality. Away from criticality, the tail was observed to be Gaussian. In this work, we generalizethe flat redistribution within the Affine Wealth Model to allow for an essentially arbitrary redistri-bution policy. We show that the exponential tail observed near criticality in prior work is in fact aspecial case of a much broader class of critical, slower-than-Gaussian decays that depend sensitivelyon the corresponding asymptotic behavior of the progressive redistribution model used. We therebydemonstrate that the functional form of the tail of the wealth distribution of a near-critical society isnot universal in nature, but rather is entirely determined by the specifics of public policy decisions.This is significant because most major economies today are observed to be near-critical.
Key words.
Distribution Theory, Econophysics, Kinetic Theory, Pareto Distribution, StatisticalMechanics, Wealth Distributions.
AMS subject classifications.
1. Introduction.1.1. Motivation.
The search for a universal form for the distribution of wealthdates back over a century to the pioneering work of Vilfredo Pareto, who first positedthat wealth distribution tails are decaying power laws [16, 17]. While this problemmay seem like a simple matter of data-fitting, modern work on the subject has becomevastly more complicated for at least two key reasons.The first problem is that it is no longer sufficient to fit wealth distribution tailsto particular functional forms, without some microscopic model to explain the originof those forms. Studies over the last two decades have focused on the constructionof simple models of binary transactions that can account for the form of empiricalwealth distributions [1, 10, 14]. Relating those to agent density functions and othermacroscopic observable quantities then requires advanced techniques of probabilitytheory and statistical physics [5].The second problem has to do with the paucity of wealth data. Only about asixth of the world’s countries collect reliable wealth data on their household surveys.Moreover, studies of the asymptotic behavior of the tail of the wealth distribution arenecessarily focused on obfuscated data due to a small minority of households reportingtheir wealth. For example, to protect anonymity, the US Survey of Consumer Financedoes not list the wealth of any household earning more than $100 million and manyother countries have followed suit [15].Thus, the problem of determining how the tail of a distribution of wealth behavesis still very much an open question. In this work, we demonstrate that attemptsto isolate the tail of the wealth distribution for study are, by nature, problematic.This is, in part, because the transport equations governing wealth distribution areintegrodifferential – and hence nonlocal – in nature [5, 14]. What is happening on thetail both determines and is determined by what is happening in bulk. Moreover, the ∗ Department of Mathematics, Tufts University., Medford, MA ([email protected]). † Department of Mathematics, Tufts University, Medford, MA ([email protected]).1 a r X i v : . [ q -f i n . GN ] J un S. L. POLK AND B. M. BOGHOSIAN assumption that there exists a universal form for the tail of wealth distributions is validonly when those distributions are far from a certain critical point marking the onsetof wealth condensation. Closer to that critical point, the form of the tail is subjectto the minutiae of national redistribution policies, rather than to any universal lawof wealth distribution. Because many of the largest economies in the world today lienear this critical point, it follows that one should not expect their wealth distributiontails to have a universal form.There are lasting implications of this work related to how wealth distributions areviewed as economic objects. This work implies that one cannot compare the distribu-tions of any two societies without considering the policy decisions of months, years,and decades prior. This work emphasizes the extreme importance of redistribution indetermining the form of the tail of the wealth distribution. Our work shows that evenminor policy changes can be extremely influential in large-wealth asymptotics, andindeed that a better approach to modeling wealth distributions would be to considerredistribution policy as the key driving entity determining the form of the wealthdistribution tail.
The model that we use in this work is an exampleof an asset-exchange model , first introduced in the 1980s [1], and first analyzed usingmethods of statistical physics in the 1990s [13, 12]. These models posit simple binary,stochastic transactions between randomly chosen pairs of agents. Our model is bestunderstood as the result of a historical sequence of such models leading up to it.The Yard Sale Model – proposed by Chakraborti in 2002 [10] – is an asset exchangemodel in which the transferred wealth is proportional to the wealth of the poorer agentin a pairwise transaction. The small, positive proportionality captures the plausiblefact that agents tend not to stake a large fraction of their total wealth in a singletransaction, which models a kind of risk-aversion. Remarkably, even when the winnerof a transaction is chosen with even odds in this model, wealth accumulates in thepossession of a single agent, whom we call the oligarch . This phenomenon of a finitefraction of societal wealth belonging to a vanishingly small fraction of agents wascalled wealth condensation by Bouchaud and Mezard in 2000 [8], and subsequentlystudied further by Burda et al. [9]. Chakraborti’s result may seem counter-intuitivebecause one would expect that a system relying on a fair coin to determine the winningagent should not confer an advantage to any one economic agent. From an economicperspective, this result is very Keynesian in that it suggests that market forces areunstable at their core and require some level of exogenous redistribution to providestability.In 2014, a Boltzmann equation was derived for the general Yard Sale Model [4].It was also shown that this equation reduces to a nonlinear integrodifferential Fokker-Planck equation, similar to the sort used in plasma kinetic theory, where the weak-transaction limit is analogous to the weak-collision limit [19]. Later in 2014, thissame universal Fokker-Planck equation was shown to be derivable by means of astochastic process [3]. The Yard Sale Model was then extended to include a flatredistribution scheme, wherein every economic agent pays an amount proportionalto his or her wealth and receives a benefit proportional to the average wealth in theeconomy . This work showed that the oligarchical time-asymptotic state describedby Chakraborti is completely mitigated under even as simple a redistribution scheme Equivalently stated, each economic agent is moved a certain fraction of the way toward themean. Those below the mean move upward, while those above the mean move downward. Theprocess pays for itself, as global wealth is conserved.HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS
Extended Yard Sale Model (EYSM) [5]. The work that introduced the EYSM alsointroduced the concept of wealth condensation criticality : This is a state where theredistribution parameter is equal to the WAA parameter. It was shown that theoligarchical share of wealth depends sensitively on these two parameters, as will bediscussed in more detail later in this work. For now, suffice it to say that supercriticalvalues of the WAA parameter – that is, values above the critical value – will resultin a partial oligarchy; subcritical values of this parameter will result in no partialoligarchy at all.Both the Yard Sale Model and its extensions assume that agent density has sup-port contained in the positive real numbers so that negative wealth is not possi-ble by construction. In 2016, however, 10.9 % of households in the United Stateswere estimated to have negative wealth (their liabilities outweighed their assets) [15].Therefore, the addition of negative wealth was seen as an important generalizationthat needed to be made to the Extended Yard Sale Model. In 2019, the Affine WealthModel (AWM) was introduced [14]. The AWM assumes that there is some fixed, max-imum amount of debt in an economy, and modifies the Extended Yard Sale Modelaccordingly. When the AWM was fit to the Survey of Consumer Finances data onwealth distributions with the Forbes 400 between 1989 and 2016, it was highly suc-cessful at modeling the United States wealth distribution with an average point-wiseerror of less than or equal to 0.16% for each fitting. In this work, we will extend theflat redistribution that was assumed in this work, in order to examine its effects onthe phenomenology of the AWM at large wealth.
The primary goal of this study is to consider the implicationsof a more general, possibly non-constant, redistribution scheme to the distributionof large wealth in the context of the above-mentioned models. We will begin byexamining the properties of the asymptotic solution to the steady-state Extended YardSale Model’s Fokker-Planck equation under general redistribution and will discuss theramifications of its behavior at large wealth. We will then generalize the asymptoticsolution obtained from the Extended Yard Sale Model to include the possibility ofnegative wealth, and thereby obtain results for the AWM.As mentioned above, prior work has demonstrated that when asymptotically fi-nite redistribution functions tend toward the WAA parameter, the oligarchical shareof wealth exhibits a phase transition [11] In this work we show that exactly how theredistribution function approaches the WAA parameter in the limit of large wealthdetermines the nature of the tail of the distribution of wealth in a near-critical society.In particular, small alterations in redistribution policy can radically change the na-ture of a distribution of wealth near criticality. In statistical physics, it is well knownthat the asymptotic behavior of distributions is extremely sensitive to model param-eters near criticality, and this work provides the equivalent observation for wealthdistribution models.
S. L. POLK AND B. M. BOGHOSIAN
Additionally, by solving the “inverse problem” of fitting data obtained from theEuropean Central Bank (ECB) to the AWM with constant redistribution parameters,we demonstrate that all of the fourteen ECB countries that we analyzed are near-critical. This implies that the tails of these countries’ wealth distributions are proneto an extreme sensitivity to the policy decisions made in that society. This stronglysuggests that minute details of those countries’ redistribution policies are more impor-tant to the shape of their wealth distributions than any universal economic principles.
In Section 2, we review notation and derive thenonlinear, integrodifferential Fokker-Planck equation for the EYSM with general re-distribution. We then solve for the functional form of the tail of the wealth distributionof the EYSM with general redistribution.In Section 3, we generalize the asymptotic form for the tail of the solution of theEYSM with general redistribution to include the possibility of negative wealth. Wethen show that the phase transition that occurs in the EYSM with general redistribu-tion also occurs in the AWM with general redistribution, and we review the derivationof the oligarchical fraction of wealth.In Section 4, we investigate the inverse problem associated with the models inSection 3. We assume knowledge of the large-wealth, steady-state agent density func-tion and solve for the redistribution function corresponding to that distribution ofwealth. We then examine possible functional forms for the decay of the tail of thewealth distribution in the case of progressive redistribution. We observe a sensitivedependence of the distribution of large wealth to the form of the redistribution func-tion in near-critical economies. Finally, we fit the flat-redistribution AWM to ECBwealth data for the fourteen countries that it serves, and demonstrate that all arenear-critical.
2. General redistribution in the Extended Yard Sale Model.2.1. Notation and the steady-state Fokker-Planck equation.
In this sec-tion, we will begin by describing the Fokker-Planck equation [18] for the ExtendedYard Sale Model (EYSM) [5]. A Fokker-Planck equation is a partial differential equa-tion that describes time evolution of a distribution influenced by drag and randomforces. In any asset exchange model, N economic agents engage in binary transactionsin which wealth is transferred. We can describe a wealth distribution in the contextof the EYSM through the use of the agent density function, P ( w, t ). In this section,we will assume all agents have nonnegative wealth, which requires P ( w, t ) to havesupport [0 , ∞ ). We will relax this assumption in Section 3.We define P ( w, t ) to be a distribution such that (cid:82) ba dw P ( w, t ) describes thenumber of economic agents with wealth between a and b at time t . It follows that (cid:82) ba dw P ( w, t ) w describes the total wealth of those agents. Hence, the total populationand total wealth can be derived from the agent density function: N := (cid:90) ∞ dw P ( w, t ) W := (cid:90) ∞ dw P ( w, t ) w. In general, population and wealth are conserved by variations of the Yard SaleModel [5], which is why we do not attach time-dependence to N and W . We let µ = W/N be the average wealth.
HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS A ( w, t ) : = 1 N (cid:90) ∞ w dx P ( x, t )(2.1) L ( w, t ) : = 1 W (cid:90) w dx P ( x, t ) x (2.2) B ( w, t ) : = 1 N (cid:90) w dx P ( x, t ) x . (2.3)The function A ( w, t ) is the fraction of agents with wealth greater than or equal to w at time t . Similarly, L ( w, t ) is the fraction of wealth held by agents whose wealthis less than or equal to w at time t . There is no easy economic interpretation for B ( w, t ), but it will nonetheless be useful in our derivations. Because this sectionassumes nonnegative wealth, A ( w, t ) uniformly decreases as wealth increases, while L ( w, t ) and B ( w, t ) uniformly increase with w . So long as the support of P ( w, t ) is asubset of the nonnegative real numbers, the range of A ( w, t ) and L ( w, t ) is [0 , ∂P∂t = − ∂∂w (cid:20) χ ( µ − w ) P (cid:21) + ∂∂w (cid:26) ζ (cid:20) µ (cid:18) B − w A (cid:19) + (1 − L ) w (cid:21) P (cid:27) + ∂ ∂w (cid:20)(cid:18) B + w A (cid:19) P (cid:21) . (2.4)where A , L , and B are the Pareto-Lorenz potentials defined in Equations (2.1)-(2.3) [5]. For ease of notation, we drop functional dependence, but at this point, P , A , L , and B are to be understood as functions of both wealth and time. Theparameter χ indicates the level of redistribution in an economy while ζ indicates thelevel of WAA in an economy. Thus, a higher χ would indicate a greater amount ofredistribution benefiting lower-wealth agents. On the other hand, a higher ζ wouldindicate a larger advantage held by wealthy agents in the modeled economy.Equation (2.4) assumes that the rate paid for redistribution is constant acrossthe wealth spectrum. However, wealth redistribution is typically non-constant andusually progressive. To generalize redistribution in the EYSM, we now introduce theconcept of a redistribution function χ ( w ), which is at the moment arbitrary apart fromthe assumption that χ ( w ) P ( w, t ) w is globally integrable for all t . The redistributionfunction χ ( w ) will be a function of wealth that returns the rate of redistributionpaid by an agent with wealth w . It is easy to see that the total wealth collected forredistribution at time t is given by T ( t ) := (cid:90) ∞ dx P ( x, t ) χ ( x ) x. (2.5)At time t , we assume that an economic agent receives a benefit proportional to theaverage redistribution collected at time t : T ( t ) N . Note that at any given time, all S. L. POLK AND B. M. BOGHOSIAN redistribution is distributed among agents so that wealth is conserved. That is, re-distribution moves from agents to agents, and not from agents to a government body.Thus, redistribution is considered distinct from taxation in our model. We leave theproblem of including non-redistributive taxation in asset exchange models to othersinterested in modeling the interplay between government and economic agents.Given this construction of χ ( w ), we can easily incorporate a generalization ofwealth redistribution into Equation (2.4): ∂P∂t = − ∂∂w (cid:20)(cid:18) T ( t ) N − χ ( w ) w (cid:19) P (cid:21) + ∂∂w (cid:26) ζ (cid:20) µ (cid:18) B − w A (cid:19) + (1 − L ) w (cid:21) P (cid:27) + ∂ ∂w (cid:20)(cid:18) B + w A (cid:19) P (cid:21) . (2.6)Conservation of population and total wealth in the nonredistributive terms was shownin prior work [5] while conservation of the redistributive term can be verified usingEquation (2.5). We set the time derivative of Equation (2.6) to zero and integrateonce with respect to wealth to obtain the following first-order nonlinear non-localordinary differential equation describing the steady-state wealth distribution: ddw (cid:20)(cid:18) B + w A (cid:19) P (cid:21) = (cid:26) TN − χ ( w ) w − ζ (cid:20) µ (cid:18) B − w A (cid:19) + (1 − L ) w (cid:21)(cid:27) P. (2.7)Note that in Equation (2.7), we have dropped dependence on time and have used totalderivatives with respect to wealth. We will henceforth follow this precedent becausewe will only be dealing with steady-state wealth distributions for the duration of thiswork. We now establishnotation that will be used extensively throughout this work. We will use the notation g ( w ) (cid:28) h ( w ) to mean lim w →∞ g ( w ) h ( w ) = 0 . (2.8)An equivalent notation often used for this functional relationship is g ( w ) = o [ h ( w )].Furthermore, we will say that g ( w ) ≈ h ( w ) iflim w →∞ g ( w ) h ( w ) = 1 . (2.9)We now will provide a set of assumptions that will enable us to approximate thelarge-wealth behavior of agent density. Using the notation defined in Equation (2.8),we will need the a priori assumption that the redistribution function χ ( w ) satisfies dχdw (cid:28) w [ χ ( w ) + αw + β ] for any real constants α and β . Note that this condition is very general and functionalforms for χ ( w ) ranging from arbitrary polynomials to the exponential of arbitrarypolynomials will satisfy it. HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ) , (2.10)where f ( w ) is a twice-differentiable, asymptotically monotone function defined on( M, ∞ )—for some M > f (cid:48) ( w ) > f (cid:48) ( w ) (cid:29) (cid:112) f (cid:48)(cid:48) ( w )(2.12) e f ( w ) (cid:29) w [ f (cid:48) ( w )] , (2.13)We assume that C is a positive constant of integration [5]. We will use only redistri-bution functions χ ( w ) for which Equations (2.11)-(2.13). These constraints are verylax and are satisfied, inter alia, by all functions of the form f ( w ) ≈ w p log q ( w ) , where either p > q ∈ R or p = 0 and q >
1. This is shown in Appendix A.The function Ξ( w ) given in Equation (2.10) is a generalized distribution thatwas introduced in prior work to represent the oligarchical fraction of wealth in aneconomy [11, 6]. There is a broad literature on the behavior of Ξ( w ), which we leaveto the reader to investigate further. Briefly, however, Ξ( w ) satisfies (cid:90) ∞ dw Ξ( w ) = 0 (cid:90) ∞ dw Ξ( w ) w = 1 (cid:90) ba dw Ξ( w ) w = 0for any a, b ∈ R . The constant c is a number between 0 and 1 representing the fractionof wealth in the possession of an oligarch. This constant will be investigated furtherin Section 2.2.2. In this section, we will de-rive an approximate form for f ( w ) based on the assumptions given in Equations(2.11)-(2.13). Prior work leads us to posit that it is the behavior of redistribution atlarge wealth which enables or inhibits oligarchy [7, 5, 11]. By better understandingthe functional form of agent density for the wealthiest agents in the distribution, wehope to better understand which redistribution policies are sufficient to preclude oli-garchy. The following lemma is necessary to make the approximations which serveas the foundation for this work. It can easily be proven through Equation (2.9), anapplication of L’Hopital’s Rule, and the consequences of our assumptions, which areprovided in Appendix B. Lemma
Under the assumptions listed in Equations (2.11)-(2.13), whenwealth is sufficiently large and m ≥ , (cid:90) ∞ w dx x m exp[ − f ( w )] ≈ w m f (cid:48) ( w ) exp[ − f ( w )] . S. L. POLK AND B. M. BOGHOSIAN
Note that we can define L ( w ) and B ( w ) in terms of integrals that are consideredby Lemma 2.1. In particular, at equilibrium, L ( w ) = L ∞ − W (cid:90) ∞ w dx P ( x ) xB ( w ) = B ∞ − N (cid:90) ∞ w dx P ( x ) x , where L ∞ and B ∞ are the complete first and second moments of P ( w ) with respectto w . Prior work has shown these to be finite numbers [5]. Intuitively, one wouldexpect L ∞ to be 1 because W is defined to be the first complete moment of P withrespect to wealth. In fact, this reasoning holds even for the asymptotic behaviorof A ( w ), and it can be proven that A ( w ) tends toward zero as w → ∞ . However,numerical simulations and analytic studies have shown that this orderly convergencedoes not hold for L ( w ) [6, 5, 11]. It has been shown that, for asymptotically constantredistribution functions χ ( w ) ≈ χ ∞ > L ∞ : = lim w →∞ L ( w ) = (cid:40) ζ ≤ χ ∞ χ ∞ ζ if ζ > χ ∞ (2.14)[5, 11]. This phenomenon was shown to be due to a second-order phase transitionobserved at criticality—a state defined by χ ∞ = ζ —that is due to Ξ( w ) [6, 11].Prior work observed that if ζ > χ ∞ – a state called supercritical – there is a partialoligarch with fraction of total wealth c = 1 − χ ∞ ζ . Conversely, if ζ < χ ∞ —a statecalled subcritical —there is no partial oligarchy whatsoever ( c = 0). This relationshipbetween redistribution and WAA explains the duality exhibited in L ∞ in Equation(2.14). In both subcritical and supercritical distributions, the distribution of largewealth is observed to be Gaussian. However, if χ ∞ = ζ – a state called critical – theoligarchical fraction of wealth drops to zero and the distribution of large wealth isobserved to be exponential [5, 11].Reducing the steady-state Fokker-Planck equation requires significant algebra,which is provided in Appendix C. The final result is that Equation (2.7) reduces to f ( w ) ≈ B ∞ (cid:90) w dx χ ( x ) x + ζ (1 − L ∞ )2 B ∞ w + 2 ζB ∞ − TN µB ∞ µ w. (2.15)The lower limit of integration in the redistributive term of Equation (2.15) is omittedbecause it is a subdominant constant of integration. Note that for any given redistri-bution function χ ( w ) and WAA parameter ζ satisfying our assumptions, we can useEquation (2.15) to find the distribution of large wealth. We emphasize that this formof f ( w ) is both an extension and corroboration of prior research on the EYSM atlarge wealth [5]. In that work, the steady-state Fokker-Planck equation’s asymptoticsolution was found to be f ( w ) ≈ | χ − ζ | B ∞ w + 2 ζB ∞ − χµ B ∞ µ w, (2.16)which is derivable from Equations (2.14) and (2.15) if redistribution is assumed to beconstant [5]. Importantly, Equation (2.15) implies that so long as the wealth limit of χ ( w ) is greater than ζ , oligarchy is impossible and L ∞ = 1. HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS
3. General Redistribution in the Affine Wealth Model.
In Section 2, wesolved for the large-wealth distribution of agent density under the assumption thatwealth is nonnegative. However, economic agents with negative wealth are widelyobserved in real-world data. For example, in 2016, 10.9 % of the population of theUnited States was estimated to have negative wealth [15]. In this section, we extendour earlier generalization of redistribution in the EYSM to a generalization of redis-tribution in the Affine Wealth Model (AWM). We will use the work of Section 2 asthe basis for this extension.The AWM is a recently-introduced Asset Exchange Model that allows for non-negative wealth. The AWM has been highly successful at modeling empirical wealthdata [14]. It is based on the EYSM, but allows the support of the agent density func-tion to be contained in [ − ∆ , ∞ ), where ∆ ≥ P ( w ) will denote the steady-state agent densityfunction explored in Section 2 and P ( w ) will denote the steady-state “shifted wealth”agent density function of the AWM. We assume that P ( w ) has support containedwithin [ − ∆ , ∞ ). The following algebraic manipulation can be easily observed, linking P ( w ) and ¯ P ( w ). P ( w ) := ¯ P ( w + ∆) . (3.1)If we make the same a posteriori assumptions on agent density that we made inSection 2.2.1—given in Equations (2.11)-(2.13)—we arrive at an approximate formfor ¯ f ( w ) ≈ − log[ ¯ P ( w )], given by Equation (2.15). By Equations (2.15) and (3.1), thedistribution of large wealth is given by f ( w ) ≈ − log[ P ( w )] − log( C ) ≈ ¯ f ( w + ∆) − log( C ) ≈ B ∞ (cid:90) w +∆ dx χ ( x ) x + ζ (1 − L ∞ )2 ¯ B ∞ ( w + ∆) + 2 ζ ¯ B ∞ − TN ¯ µ ¯ B ∞ ¯ µ ( w + ∆) − log( C ) . Next, we will expand the quadratic term and group by the power in wealth:= 1¯ B ∞ (cid:90) w +∆ dx χ ( x ) x + ζ (1 − L ∞ )2 ¯ B ∞ w + 2 ζ ¯ B ∞ − TN ¯ µ + ζ ∆¯ µ (1 − L ∞ )¯ B ∞ ¯ µ w − log( C ) , (3.2) = ¯ f ( w ) + 1¯ B ∞ (cid:90) w +∆ w dx χ ( x ) x + ζ ∆(1 − L ∞ )¯ B ∞ w (3.3)0 S. L. POLK AND B. M. BOGHOSIAN where, without loss of generality, the constant term has been absorbed into the con-stant of integration C . Hence, for any redistribution function χ ( w ) and WAA param-eter ζ , there exists a function f ( w ) such that the agent density decays according to P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ). We provide the AWM distribution at large wealth interms of the EYSM distribution in Equation (3.3) with correction terms.In Section 2.2.2, we reviewed the results of prior work that showed the existence ofa second-order phase transition in the EYSM. This phase transition in the oligarchicalshare of wealth occurred whenever the limit of the redistribution χ ∞ is less than ζ [7, 5, 11]. An analogous result holds for the AWM. We will spend the duration ofthis section reviewing the derivation of the oligarchical share of wealth in the AWM.For ease of notation, we introduce the parameter λ ≥
0, which is defined implicitlyby ∆ = λ ¯ µ. In general, a larger λ corresponds to a larger maximum value of debt in the model.Let ¯ A ( w ) be the EYSM Pareto-Lorenz Potential defined in Equation (2.1). Let ¯ L ( w )be the Pareto-Lorenz potential defined in Equation (2.2) for the EYSM, and let L ( w )be its AWM equivalent. These two integral operators can be shown to be related inthe following way: L ( w ) = (1 + λ ) ¯ L ( w + ∆) − λ [1 − ¯ A ( w + ∆)] . [14] Note that ¯ A ( w + ∆) and ¯ L ( w + ∆) are two of the Pareto potentials in the EYSM.Then as w → ∞ , ¯ A ( w + ∆) → L ( w + ∆) → ¯ L ∞ : the asymptotic limit of ¯ L ( w )defined in Equation (2.14). Hence, L ∞ = lim w →∞ (cid:26) (1 + λ ) ¯ L ( w + ∆) − λ [1 − ¯ A ( w + ∆)] (cid:27) = (1 + λ ) ¯ L ∞ − λ. (3.4)By Equation (3.4), the fraction of wealth held by the oligarch in the AWM isclosely related to that which was introduced in the EYSM. Moreover, due to theinclusion of ¯ L ∞ in Equation (3.4), the AWM exhibits a similar phase transition tothat of the EYSM. Suppose that χ ∞ ≥ ζ , as would be the case in a subcritical orcritical economy. By Equations (2.14) and (3.4), L ∞ = (1 + λ ) × − λ = 1 . This implies that the oligarchical share of wealth is zero. Suppose that χ ∞ < ζ , aswould be the case in an EYSM supercritical economy. By Equations (2.14) and (3.4), L ∞ = (1 + λ ) χ ∞ ζ − λ. Hence, the fraction of wealth held by the oligarch is equal to c = 1 − L ∞ = (1 + λ ) (cid:18) − χ ∞ ζ (cid:19) . (3.5)Therefore, the fraction of wealth held by the partial oligarch is magnified by theamount of negative wealth in a society. These results are summarized in the followingequation: L ∞ = (cid:40) χ ∞ ≥ ζ (1 + λ ) χ ∞ ζ − λ χ ∞ < ζ . (3.6) HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS λ = 0, as would be the case for nonnegative wealth, Equation (3.6)reverts back to the asymptotics of ¯ L ( w ) in the EYSM. Hence, this argument trulydoes generalize the EYSM with general redistribution to include the possibility ofnegative wealth.
4. Inverse problem.4.1. Redistribution as a function of large-wealth agent density.
Afterderiving Equation (3.2), we observed that for any redistribution function χ ( w ) andWAA parameter ζ , there exists a function f ( w ) such that agent density decays accord-ing to P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ) where c depends on the limit of χ ( w ) relative to ζ .It is worth noting that the converse of this statement may not be true. In this section,we will delve into the inverse problem to Section 3. In particular, given that policy-makers would like the distribution of large wealth to obey P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ),what redistribution policy should they follow? Is the distribution they have in mindeven possible within the context of the AWM? In this section, we aim to provide aformal answer to these questions.Throughout this section, we will assume knowledge of a function f ( w )—twicedifferentiable on ( M, ∞ ) for some M > − ∆—such that agent density at large wealthis of the form P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ). Thus, we allow for the possibility ofoligarchy in our a priori assumption on P ( w ). However, because we are primarilyconcerned with the distribution of large wealth and not that of oligarchs, we willnot delve into the deep and interesting literature on Ξ( w ). We refer any readersinterested in the oligarch’s contribution to the distribution of wealth to prior workon this subject [7, 11]. For now, the constant c = 1 − L ∞ can be thought of as aparameter to be tuned by legislators when drafting policy. Many of the followingderivations will be analogous to those in Section 2. The important distinction isthat in Section 2, we assumed knowledge of χ ( w ) and an a priori asymptotic form P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ) for some function f ( w ) that satisfies Equations (2.11)-(2.13). In this section, we assume knowledge of a P ( w ) satisfying Equations (2.11)-(2.13) and hence the asymptotic form P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ), but make the apriori assumption that there is a χ ( w ) that will produce the asymptotic distributionof P ( w ).As in Section 2.1, we set the time-derivative of the EYSM’s Fokker-Planck equa-tion with general redistribution—Equation (2.6)—equal to zero and integrate oncewith respect to wealth to obtain Equation (2.7). Note that Lemma 2.1 is still appli-cable for asymptotic approximations, as any EYSM agent density function that wewill consider satisfies the conditions stated in Equations (2.11)-(2.13) by assumption.Then the approximations that were made in Appendix C to derive the asymptoticform of ¯ f ( w ) from the steady-state EYSM Fokker-Planck equation are still valid, andwe arrive at Equation (2.15), as in Section 2.2.2. We then can apply the transforma-tion discussed in Section 3 to obtain a form for f ( w ): the negative logarithm of theAWM agent density function. At this point, we rearrange Equation (3.2) to find thatthe redistribution function χ ( w ) must satisfy (cid:90) w +∆ dx χ ( x ) x = ¯ B ∞ ¯ f ( w ) + ζ (2 ¯ L ∞ − w + 2 ζ ¯ B ∞ − TN ¯ µ + ζ ∆¯ µ (1 − L ∞ )¯ µ w if it exists for a given distribution P ( w ). Applying the Fundamental Theorem of2 S. L. POLK AND B. M. BOGHOSIAN
Calculus and dividing by w , we find that χ ( w ) must have the asymptotic form χ ( w + ∆) = ζ (2 ¯ L ∞ −
1) + ¯ B ∞ ¯ f (cid:48) ( w ) w + 2 ζ ¯ B ∞ − TN ¯ µ + ζ ∆¯ µ (1 − L ∞ )¯ µ w (4.1)We can thus find χ ( w ) by inputting w − ∆ on the right hand side of Equation (4.1).We now will apply a Maclaurin expansion of χ ( w ) about ∆. χ ( w ) = ζ (2 ¯ L ∞ −
1) + ¯ B ∞ ¯ f (cid:48) ( w ) w + 2 ζ ¯ B ∞ − TN ¯ µ ¯ µ w + O (cid:18) ∆ w (cid:19) , where we have used big- O notation to refer to subdominant terms in the Taylorexpansion. We assume that w is very large so that ∆ is small relative to w . Then ∆is even smaller compared to w , and we may approximate χ ( w ) by its leading terms: χ ( w ) ≈ ζ (2 ¯ L ∞ −
1) + ¯ B ∞ ¯ f (cid:48) ( w ) w + 2 ζ ¯ B ∞ − TN ¯ µ ¯ µ w . (4.2)The reader may be concerned that χ ( w ) is defined in terms of T , which is itself afunctional of χ ( w ). We note that for all redistribution functions we consider relevant,the total redistribution will be constant. For this reason, we assume that it is anarbitrary constant for the purpose of solving Equation (4.2) and assume that its valuecan be set once χ ( w ) and P ( w ) are known.Equation (4.2) implies that we can describe asymptotic redistribution as the sumof a constant and some function of wealth. In particular, we let χ ( w ) : ≈ ζ (2 ¯ L ∞ −
1) + ι ( w ) , where ι ( w ) is defined by ι ( w ) : ≈ ¯ B ∞ ¯ f (cid:48) ( w ) w + 2 ζ ¯ B ∞ − TN ¯ µ ¯ µ w . We note that the behavior of ι ( w ) – as the sole non-constant contribution to redis-tribution at large wealth – will dictate the large-wealth behavior of redistribution.At this point, we are free to consider the implications of the derived form of χ ( w ),which is in terms of f (cid:48) ( w ). To do this, we introduce some important terminology.Assume P ( w ) ≈ Ce − f ( w ) + cW Ξ( w ), where f ( w ) is a differentiable function. Wesay that P ( w ) decays sub-quadratically if f (cid:48) ( w ) (cid:28) w . Similarly, we say that P ( w )decays quadratically if f (cid:48) ( w ) ≈ aw for some a (cid:54) = 0. Finally, we say that P ( w ) decays super-quadratically if f (cid:48) ( w ) (cid:29) w . Prior work on the AWM observed a critical relationship in the case where the redis-tribution function χ ( w ) is asymptotically constant, tending towards a limit we shallcall χ ∞ [11]. When χ ∞ < ζ , agent density was observed to decay like a Gaussian,and there was a partial oligarchy with share of wealth c = 1 − χ ∞ ζ . By contrast, when χ > ζ , agent density decayed like a Gaussian with no oligarchy whatsoever ( c = 0).However, when χ ∞ = ζ – a state introduced earlier as criticality – agent densitywas observed to decay exponentially, with no oligarch. In this subsection, we willshow that the critical exponential decay that was observed in prior work is actuallya special case of a more general family of sub-quadratic decays [11, 6]. We will prove HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS χ ( w ) ≈ ζ andno oligarchy.Suppose that f ( w ) is a twice-differentiable function on ( M, ∞ ) for some M > − ∆,that it satisfies Equations (2.11)-(2.13), and that f (cid:48) ( w ) (cid:28) w , a condition whichcorresponds to agent density decaying sub-quadratically. It is clear from Equation(4.1) that ι ( w ) → w → ∞ , so that χ ( w ) → ζ (2 ¯ L ∞ − χ ( w ) is such that all higher-order (quadratic and linear) terms are canceled in theasymptotic form for f ( w ). So, ι ( w ) contributes non-negligibly to large-wealth agentdensity. Prior work has dealt only with constant redistribution schemes. Thereforethe critical exponential distribution – the case in which χ ( w ) = χ = ζ and ¯ L ∞ = 1– seemed like the unique sub-quadratic decay satisfying these properties [11]. Theabove argument extends this idea to a family of sub-quadratic distributions. We haveassumed nothing about the existence or nonexistence of oligarchy in the sub-quadraticcase, but note that if ¯ L ∞ = 1, this argument describes redistribution functions χ ( w ),tending toward ζ as w → ∞ , that will produce other sub-quadratic decays that arenot exponential.We have shown that many sub-quadratic decays are possible by allowing non-constant redistribution, and that to attain such a distribution of large wealth, χ ( w )must tend toward ζ (2 ¯ L ∞ −
1) as w → ∞ . However, the way that χ ( w ) approachesthis limit warrants further discussion. Suppose that a government is aiming for asub-quadratic distribution of wealth C e − g ( w ) but that the limit of redistribution itaims for differs by a small margin from ζ (1 − L ∞ ). In particular, if (cid:15) ∈ R is somesmall, possibly negative constant, suppose that χ ( w ) → ζ (2 ¯ L ∞ −
1) + (cid:15) as wealthbecomes large. In this case, Equation (2.15) tells us that agent density will behavelike C e − f ( w ) + cW Ξ( w ), where f ( w ) ≈ B ∞ (cid:90) w +∆ dx (cid:20) (cid:15) x + ¯ B ∞ g (cid:48) ( x ) (cid:21) (4.3)and c is implied implicitly by c = 1 − L ∞ and Equation (3.6). We have assumedthat g (cid:48) ( w ) (cid:28) w , so Equation (4.3) will at some point be well-approximated by anorder- w term. However, there may be a section of the wealth distribution where thecontribution from g ( x ) competes with (cid:15) w if the value (cid:15) is sufficiently small.Importantly, the point at which the linear term in the integrand of Equation (4.3)dominates the g (cid:48) ( w ) term may be near the end of the wealth spectrum, where thediscretization of agent density will make it irrelevant. By continuity, there is a point w (cid:15) at which g (cid:48) ( w (cid:15) ) = (cid:15)w (cid:15) . At this point, the two terms in the integrand of Equation(4.3) become comparable, but for wealth much lower than this, P ( w ) ≈ C e − g ( w ) .Similarly, for wealth much greater than w (cid:15) , agent density will decay like a Gaussian,and the theory of prior work on criticality applies [11, 14]. However, if it is truethat N A ( w (cid:15) ) <
1, where A ( w ) is defined to be the AWM equivalent to the Pareto-Lorenz potential given in Equation (2.1), there will be no economic agents with wealthgreater than w (cid:15) . This argument shows that the discretization of agent density allowsthe limit of redistribution to differ from the WAA parameter while still attaining asub-quadratic distribution of wealth.The redistribution function corresponds to the policy choices of a society, andthis analysis shows that those choices are of the utmost importance for economiesnear criticality. The redistribution function has been broken up into its constant andnon-constant contributions. In the case of a sub-quadratic decays in agent density,the non-constant contribution ι ( w ) tends towards zero as w → ∞ . However, the4 S. L. POLK AND B. M. BOGHOSIAN argument of this section has shown that the way in which ι ( w ) tends towards zeroindicates the nature of the sub-quadratic decay which is attained. Hence, when aneconomy is near-critical, the qualitative nature of its distribution of large wealth isextremely sensitive to the trivialities of redistribution policy. The workof Section 4.2 implies that it is possible to attain any sub-quadratic tail given someredistribution function. In the literature on wealth distribution tails, many formshave been fitted to empirical data. In this section, we will provide the asymptoticredistribution function necessary to attain common distributions in the study of tailsof wealth distributions. To do this, we first convert a probability density function toan agent density function by multiplying by N . We then consider the dominant termor terms in the asymptotic form of f ( w ) ≈ − log( P ). From this, it is easy to derivethe asymptotic redistribution function from Equation (4.2). We will assume that if P has a sub-quadratic decay, then L ∞ = 1, as is the case of an exponential decay [6, 11].These results are provided in Table 1. Table 1
The redistribution functions for six classes of distributions that are commonlyobserved within the study of wealth economics.
The redistribution function χ ( w ) is derivedfrom Equation (4.2). We have used in the case of the Exponential distribution that the Dw termmust cancel due to Equation (2.15). We also assume for higher-order Gaussian distributions that m ∈ (1 , ∞ ) . For ease of notation, we define the constant D implicitly by ι ( w ) = B ∞ f (cid:48) ( w ) w + Dw . Wealso define C to be some normalization constant and forgo the barring of B ∞ and µ . Distribution Agent Density Function Asymptotic f ( w ) χ ( w )Exponential C exp[ − λw ] λw ζ Log-normal C w exp (cid:2) − [ln( w ) − σ (cid:3) σ log( w ) ζ + Dw + B ∞ σ log( w ) w Pareto Cw α +1 ( α + 1) log( w ) ζ + Dw + ( α +1) B ∞ w Inverse-Gamma Cw − ( α +1) exp (cid:2) − βw (cid:3) ( α + 1) log( w ) + βw ζ + Dw + ( α +1) B ∞ − βB ∞ w Gaussian C exp (cid:2) − ( w − µ ) σ (cid:3) w σ lim w →∞ χ ( w ) (cid:54) = ζ Higher-OrderGaussian C exp (cid:2) − ( w − µ ) m σ m (cid:3) w m σ m mB ∞ σ m w m − The first four rows of Table 1 consist of sub-quadratic decays. Notably, each ofthe redistribution functions necessary to attain these wealth distribution tails differfrom one another by solely a sub-dominant term on the order of w or w . This resultis all the more evident when comparing the Pareto and Inverse-Gamma distributions’redistribution functions at large wealth. The necessary redistribution functions forthese two different decays vary by a factor of solely − ¯ B ∞ βw . This result emphasizes theimportance of the results in Section 4.2. The minute details of redistribution policyhave a dynamic effect on the shape of wealth distributions in their tails when nearwealth condensation criticality. In Section 4.1, weshowed that when an economy is near-critical, the minor details of its redistributionpolicy govern its wealth distribution’s tail’s behavior. In this section, we will present
HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS L ( F ) is a parametric plot of L ( w )against F ( w ) := 1 − A ( w ). This is a curve in the unit square, parametrized by thewealth w , that can be shown to be concave up and lie below the diagonal. A point( f, l ) on the Lorenz curve tells us that a fraction l of wealth is held by a fraction f of economic agents. The farther the Lorenz curve is from the diagonal of the unitsquare, the more inequality exists in an economy. Inequality can be quantified usingthe Gini coefficient G , which is defined to be two times the area under the Lorenzcurve. A Gini coefficient of 0 would therefore represent a total oligarchy and a Ginicoefficient of 1 would represent a uniform distribution of wealth.First, we set χ ( w ) = χ to be a constant function in this section. Thus, ourparameter space is θ = { χ, ζ, λ } . Let L ( F ) be the empirical Lorenz curve and L θ ( F )is the theoretical Lorenz curve for the theoretical (model) Lorenz curve of the agentdensity function given by the AWM with parameters θ := { χ, ζ, λ } [14]. We definethe discrepancy by J ( θ ) := (cid:90) dF |L ( F ) − L θ ( F ) | . (4.4)Thus, J ( θ ) is the L norm of the difference between the empirical Lorenz curve andthat of the AWM with parameter choices θ . The fittings to ECB data were performedby minimizing J ( θ ) over θ . There are no guarantees for the concavity of J ( θ ), so weemployed a global numerical search for the optimal parameters. The optimal valuesfor χ , ζ , and λ are given in Table 2. We let G fit refer to the Gini coefficient of theAWM-fitted Lorenz curve. Table 2
Optimal redistribution, WAA, and negative wealth parameters for the AWM whenfitted to fourteen ECB countries’ empirical wealth data.
The values χ opt , ζ opt , and λ opt are defined to be the optimal parameters for the AWM fitting to a given country’s ECB wealth data. G fit is the Gini coefficient of the corresponding wealth distribution, as obtained by the Affine WealthModel [14]. Country χ opt ζ opt λ opt G fit Austria 0.156 0.182 0.185 0.763Belgium 1.406 1.514 0.577 0.589Cyprus 0.164 0.190 0.096 0.690Germany 0.162 0.184 0.199 0.759Spain 1.568 1.728 0.502 0.568Finland 0.972 1.000 0.639 0.665France 0.556 0.608 0.286 0.673Greece 1.944 2.000 0.650 0.553Italy 1.194 1.300 0.502 0.601Lithuania 0.896 1.066 0.425 0.658Malta 1.154 1.348 0.377 0.583Netherlands 1.676 1.516 0.992 0.647Portugal 0.564 0.678 0.309 0.672Slovenia 1.978 1.998 0.618 0.529
It is notable that our data fittings of the AWM found all fourteen Europeancountries to be near-critical (Fig 1). The theoretical work in Section 4.2 showedthat when an economy is near-critical, the qualitative nature of the distribution oflarge wealth depends sensitively on redistribution policy. We have shown that the6
S. L. POLK AND B. M. BOGHOSIAN exact nature of the distribution of large wealth in a near-critical economy dependssensitively on the minute details of public policy, which are modeled by ι ( w ). Thus,it is possible that each of these countries has a qualitatively different sub-quadraticdecay in agent density at large wealth. opt op t Fittings of Affine Wealth Model with constant redistribution
AT BECY DE ESFIFR GRITLU MT NLPT SI
Fig. 1 . Plots of optimal redistribution parameters ( χ opt ) of against optimal WAAparameters ( ζ opt ) for fourteen ECB countries. For some of the results of our fittings, the error was so small that the Lorenz curvegiven by our model was difficult to discriminate from that of the empirical data. Forthis reason, we will consider the performance of our model in terms of local error aswell. We define the local error as the length of a line segment connecting the empiricaldata point ( f j , l j ) to the model Lorenz curve, constructed so as to be perpendicularto the latter. This section’s models assumed that redistribution was constant in theAWM. Therefore, if the limit of redistribution was not exactly equal to the WAAparameter, the distribution of large wealth was assumed to be Gaussian. Despitethe small point-wise error of these fittings, it is notable that for many countries, thevast majority of point-wise error in the fittings occurs in the tail. Four excellentexamples of this relatively large error in the tail are provided in Fig 2. We conjecturethat the point-wise error in the tail of the distribution is explained by the state ofnear-criticality and the assumption of a Gaussian distribution at large wealth. Ourwork in Section 4.2 shows that because these countries are near-critical, there couldbe a non-exponential, sub-quadratic decay which describes the tail of these countriesdistributions better than a Gaussian. HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS (a) (b)(c) (d) Fig. 2 . Optimal fits of the AWM to ECB data for four countries.
For each country, wedetermined the parameter ( χ, ζ, λ ) that minimizes the L -norm of the difference between empiricaland model Lorenz curves. Pointwise error is plotted within each figure. The fraction of oligarchicalwealth can be estimated by Equation (3.6). Notably, the local error is large in the tail, suggestingthe presence of a non-exponential sub-quadratic decay. (a): Belgium. (b): Luxembourg. (c): Malta.(d): Portugal.
5. Conclusion.
This brings us to the overall conclusion of this work: that thereis no universal form for most real-world economies’ wealth distribution tails. We haveshown that the nature of the asymptotic solution to the Fokker-Planck equation gov-erning the AWM sensitively depends on one’s choice in redistribution policy when aneconomy is near-critical (Section 4.2). When the AWM was fit to the wealth data offourteen European Union countries, we found that each one was near-critical (Table2). We conclude that the popular question of whether wealth decays like a Paretodistribution or an exponential distribution cannot be answered without first consid-ering the policies of the country from which data was collected. This implies thatthere is no universal form for wealth distributions, at least at large wealth. Becauseof the exhibited sensitivity that a distribution of wealth has to the particularities ofredistribution policy, we suggest a reframing of how the distribution of large wealth isstudied. In particular, rather than fitting distributions to wealth data and observing8
S. L. POLK AND B. M. BOGHOSIAN how accurate those fittings may be, we suggest an emphasis on studying redistribu-tion policy and its effects on the entire wealth distribution. This approach wouldmore readily capture the integrodifferential nature of the equations governing wealthdistributions [14]. The bulk of econophysics research, in tandem with the work de-scribed in our research, shows that this is a more scientific and well-posed approachto understanding a distribution of wealth.In this work, we have extended the Extended Yard Sale and Affine Wealth Mod-els by generalizing redistribution to be a nearly arbitrary function of wealth [5, 14].We showed that every sub-quadratic decay satisfying the assumptions of this work ispossible by means of a progressive redistribution function with a wealth limit withina neighborhood of the WAA parameter. This extends the notion of criticality—thephenomenon of the presence of oligarchy disappearing when constant redistributionis exactly equal to the WAA parameter—to include a plethora of sub-quadratic de-cays other than the exponential considered by prior work [5, 14]. These include thelognormal and Pareto distributions. We note that the way that the redistributionfunction tends towards its asymptote governs the distribution of wealth. This impliesthat near-critical systems are extremely sensitive to the minutiae of redistributionpolicy. Moreover, this sensitive dependence of the nature of wealth distributions im-plies that the redistribution policy decisions of a society are more indicative of thedistribution of large wealth than any underlying economic forces when that society isnear criticality.The fact that the asymptotic redistribution rate need not be exactly equal to theWAA parameter is of the utmost importance to policy decisions in global economics.We fit the AWM to empirical wealth data from fourteen European economies andfound that all lie either just above or below criticality (Fig 1). Our work impliesthat the conversation about how large wealth is distributed may be ill-posed, as thedistribution will sensitively on the policy decision specific to those countries. Thisimplies that a universal distribution of wealth like that which has been sought for fromPareto to Piketty seeks is likely a chimera. We emphasize that to understand a near-critical wealth distribution, one must thoroughly analyze large wealth redistributionpolicies that vary across societies.
6. Acknowledgements.
For part of the time spent on this work, B.M.B. wasvisiting the Research and Training Center of the Central Bank of Armenia in 2017.The authors gratefully acknowledge the support and hospitality of the Central Bankof Armenia. We also would like to thank Hongyan Wang, Chengli Li, and Jie Li fortheir help with the model fittings presented in this work.
REFERENCES[1]
J. Angle , The surplus theory of social stratification and the size distribution of personal wealth ,Social Forces, 65 (1986), pp. 293–326.[2]
J. Baldwin , Nobody Knows My Name: More Notes of a Native Son , Dial Press, 1961.[3]
B. Boghosian , Fokker–planck description of wealth dynamics and the origin of pareto’s law ,International Journal of Modern Physics C, 25 (2014), p. 1441008.[4]
B. M. Boghosian , Kinetics of wealth and the pareto law , Physical Review E, 89 (2014),p. 042804.[5]
B. M. Boghosian, A. Devitt-Lee, M. Johnson, J. Li, J. A. Marcq, and H. Wang , Oli-garchy as a phase transition: The effect of wealth-attained advantage in a fokker–planckdescription of asset exchange , Physica A: Statistical Mechanics and its Applications, 476(2017), pp. 15–37.[6]
B. M. Boghosian, A. Devitt-Lee, J. Li, J. A. Marcq, and H. Wang , Describing realisticwealth distributions with the extended yard-sale model of asset exchange , arXiv preprintHE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS arXiv:1604.02370, (2016).[7] B. M. Boghosian, M. Johnson, and J. A. Marcq , An h theorem for boltzmann?s equation forthe yard-sale model of asset exchange , Journal of Statistical Physics, 161 (2015), pp. 1339–1350.[8]
J.-P. Bouchaud and M. M´ezard , Wealth condensation in a simple model of economy , PhysicaA: Statistical Mechanics and its Applications, 282 (2000), pp. 536–545.[9]
Z. Burda, D. Johnston, J. Jurkiewicz, M. Kami´nski, M. A. Nowak, G. Papp, and I. Zahed , Wealth condensation in pareto macroeconomies , Physical Review E, 65 (2002), p. 026102.[10]
A. Chakraborti , Distributions of money in model markets of economy , International Journalof Modern Physics C, 13 (2002), pp. 1315–1321.[11]
A. Devitt-Lee, H. Wang, J. Li, and B. Boghosian , A nonstandard description of wealthconcentration in large-scale economies , SIAM Journal on Applied Mathematics, 78 (2018),pp. 996–1008.[12]
A. Dr˘agulescu and V. M. Yakovenko , Exponential and power-law probability distributionsof wealth and income in the united kingdom and the united states , Physica A: StatisticalMechanics and its Applications, 299 (2001), pp. 213–221.[13]
S. Ispolatov, P. L. Krapivsky, and S. Redner , Wealth distributions in asset exchangemodels , The European Physical Journal B-Condensed Matter and Complex Systems, 2(1998), pp. 267–276.[14]
J. Li, B. M. Boghosian, and C. Li , The affine wealth model: An agent-based model of assetexchange that allows for negative-wealth agents and its empirical validation , Physica A:Statistical Mechanics and its Applications, 516 (2019), pp. 423–442.[15]
B. of Governors of the Federal Reserve System , The survey of consumer finances , 2016.[16]
V. Pareto , Cours d’Economie Politique: Nouvelle edition par G.-H. Bousquet et G. Busino ,Librairie Droz, Geneva, 1964.[17]
V. Pareto , La Courbe de la repartition de la richesse , Imprimerie Ch. Viret-Genton, Geneva,1965.[18]
H. Risken , Fokker-planck equation , in The Fokker-Planck Equation, Springer, 1996.[19]
M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd , Fokker-planck equation for aninverse-square force , Physical Review, 107 (1957), p. 1.
Appendix A. Class of functions satisfying our assumptions.
In this section, we prove that given a function f ( w ) satisfying some loose condi-tions, the assumptions of this work will hold. Proposition
A.1. If f ( w ) ≈ w p log q ( w ) where either p > , q ∈ R or p = 0 , q > , Equations (2.11)-(2.13) are satisfied.Proof. Case 1 : Assume that p > q ∈ R .1. By assumption, f (cid:48) ( w ) ≈ ap w p − log q ( w ), which is positive for all largewealth. Therefore, Equation (2.11) holds.2. Consider the following limit:lim w →∞ f (cid:48)(cid:48) ( w )[ f (cid:48) ( w )] = 1 a lim w →∞ p ( p −
1) log ( w ) + (2 p − q log( w ) + q ( q − w p log q ( w )[ p log( w ) + q ] = p − ap lim w →∞ q ( w ) w p = 0 . Therefore, Equation (2.12) holds.0
S. L. POLK AND B. M. BOGHOSIAN
3. Consider the following limit:lim w →∞ w [ f (cid:48) ( w )] e − f ( w ) = lim w →∞ log − q ( w ) e − aw p log q ( w ) a w p − ( p log( w ) + q ) = 1 a p lim w →∞ e − aw p log q ( w ) w p − log q ( w )= 1 a p lim w →∞ w − aw p log q − ( w ) w p − log q ( w )= 0 . Therefore, Equation (2.13) holds
Case 2 : Assume that p = 0 and q >
1. Then1. By assumption, f (cid:48) ( w ) ≈ q log q − ( w ) w , which is positive for all large wealth.Thus, Equation (2.11) holds.2. Consider the following limit:lim w →∞ f (cid:48)(cid:48) ( w )[ f (cid:48) ( w )] = lim w →∞ (cid:20) q − q log q ( w ) − q − ( w ) (cid:21) = 0 . Thus, Equation (2.12) holds.3. Consider the following limit:lim w →∞ w [ f (cid:48) ( w )] e − f ( w ) = lim w →∞ w q log q − exp[ − log q ( w )]= lim w →∞ w − log q − ( w ) q log q − ( w )= 0 . Thus, Equation (2.13) holds.Therefore, our assumptions are satisfied by the following class of functions: w p log q ( w ), where either p > q ∈ R , or p = 0 and q > Appendix B. Further notes on the consequences of our asymptoticassumptions.
In this section, we derive some facts which are used throughout this work.Lemmas B.2-B.4 are consequences of our assumptions: Equations (2.11)-(2.13). Weassume that wealth is sufficiently large that these assumptions are valid.
Lemma
B.1. If lim w →∞ f ( w ) = ∞ and g (cid:48) ( w ) (cid:29) f (cid:48) ( w ) , then g ( w ) (cid:29) f ( w ) .Proof. Because g (cid:48) ( w ) (cid:29) f (cid:48) ( w ) for all positive M , however large, there exists an n M >
0, where w > n M implies that g (cid:48) ( w ) > M f (cid:48) ( w ). This implies that g ( w ) = g ( n M ) + (cid:90) wn M dx g (cid:48) ( x ) g ( w ) > g ( n M ) + M (cid:90) wn M dx f (cid:48) ( x ) g ( w ) > [ g ( n M ) − M f ( n M )] + M f ( w ) g ( w ) f ( w ) > g ( n M ) f ( w ) − M f ( n M ) f ( w ) + M. HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS n M is a constant and that f ( w ) is an infinite function,for w large, f ( n M ) f ( w ) ≈ g ( n M ) f ( w ) ≈
0. Then at large wealth, we have that there exists an n (cid:48) M > w > n (cid:48) M , it is true that g ( w ) > M f ( w ). Thus, g ( w ) (cid:29) f ( w ). Lemma
B.2.
Under the assumptions listed in Equations (2.11)-(2.13), it can beproven that exp[ f ( w )] (cid:29) w f (cid:48) ( w ) .Proof. By Equation (2.13), e f ( w ) (cid:29) w [ f (cid:48) ( w )] . Since w [ f (cid:48) ( w )] (cid:29) wf (cid:48) ( w ) , we havethrough transitivity that e f ( w ) (cid:29) wf (cid:48) ( w ) . Thus, e f ( w ) (cid:29) wf (cid:48) ( w ) f (cid:48) ( w ) e f ( w ) (cid:29) w. Note that because f (cid:48) ( w ) e f ( w ) (cid:29) w , both are necessarily infinite functions of wealth.Therefore, the assumptions of Lemma B.1 are satisfied and by integrating, we see that e f ( w ) (cid:29) w . (B.1)Taking the geometric mean of Equation (2.13) and Equation (B.1), we arrive at ourresult. e f ( w ) = (cid:112) e f ( w ) (cid:29) (cid:115) w w [ f (cid:48) ( w )] = w f (cid:48) ( w ) . Lemma
B.3.
Under the assumptions listed in Equations (2.11)-(2.13), exp[ f ( w )] (cid:29) w .Proof. By Equation (B.1) in our proof of Lemma B.2, exp[ f ( w )] (cid:29) w (cid:29) w . Lemma
B.4.
Under the assumptions listed in Equations (2.11)-(2.13), i f (cid:48) ( w ) (cid:29) w . Proof. By Lemma B.3, exp[ f ( w )] (cid:29) wf ( w ) (cid:29) log( w ) . In particular, 0 = lim w →∞ log( w ) f ( w ) = lim w →∞ w f (cid:48) ( w ) , where we have applied L’Hopital’s Rule to arrive at our conclusion. Appendix C. Reduction of the steady-state Extended Yard Sale ModelFokker-Planck equation.
In this appendix, we will derive Equation (2.15) using the assumptions providedin Equations (2.11)-(2.13). First, let us rearrange the steady-state EYSM Fokker-Planck equation with general redistribution–Equation (2.7)– ddw [log P ] = TN − χ ( w ) w − ζ (cid:2) NW (cid:0) B − w A (cid:1) + (1 − L ) w (cid:3) − wAPB + w A . (C.1)2
S. L. POLK AND B. M. BOGHOSIAN
By Lemma 2.1, we can approximate the Pareto-Lorenz potentials on ( M, ∞ ) by thefollowing functional forms: A ( w ) = 1 N (cid:90) ∞ w dx P ( x ) ≈ N f (cid:48) ( w ) P ( w )(C.2) L ( w ) = L ∞ − W (cid:90) ∞ w dx xP ( x ) ≈ L ∞ − W wf (cid:48) ( w ) P ( w )(C.3) B ( w ) = B ∞ − N (cid:90) ∞ w dx x P ( x ) ≈ B ∞ − N w f (cid:48) ( w ) P ( w ) . (C.4)The process by which Equation (C.1) reduces to Equation (2.15) is rather arduous andrequires the application of the assumptions listed in Equations (2.11)-(2.13) as wellas their consequences, which are provided in Section B. For this reason, we will provethree lemmas (Lemmas C.1-C.3). Then, in Proposition C.4, we will show that thesethree lemmas can be used to derive Equation (2.15) as a large-wealth approximationof P ( w ). Lemma
C.1.
At large wealth, B ± w A ≈ B ∞ . Proof.
Consider this quantity in the context of our asymptotic approximations ofthe Pareto-Lorenz Potentials. B ± w A ≈ B ∞ − N w f (cid:48) ( w ) P ( w ) ± N w f (cid:48) ( w ) P ( w )= B ∞ + C ( − ± N w f (cid:48) ( w ) e − f ( w ) = (cid:20) B ∞ e f ( w ) + C ( − ± N w f (cid:48) ( w ) (cid:21) e − f ( w ) ≈ B ∞ , where we have used that e f ( w ) (cid:29) w f (cid:48) ( w ) , which is true by Lemma B.2 Lemma
C.2.
At large wealth, for any constant c , wAP (cid:28) c. Proof.
We will prove the equivalent condition that c + wAP ≈ c at large wealth. c + wAP = c + CN we − f ( w ) (cid:90) w dx P ( x ) = (cid:20) ce f ( w ) + CN wA ( w ) (cid:21) e − f ( w ) Recall that A ( w ) is bounded by [0 ,
1] for all w and our assumptions implied that e f ( w ) (cid:29) w , which was proven in Lemma B.3. Then c ≤ (cid:20) ce f ( w ) + CN wA ( w ) (cid:21) e − f ( w ) ≤ (cid:20) ce f ( w ) + CN w (cid:21) e − f ( w ) ≈ c. Therefore, c + wAP ≈ c . HE NON-UNIVERSALITY OF WEALTH DISTRIBUTION TAILS Lemma
C.3.
At large wealth, ζ (cid:20) NW (cid:18) B − w A (cid:19) + (1 − L ) w (cid:21) ≈ ζ (cid:20) N B ∞ W + (1 − L ∞ ) w (cid:21) . Proof.
Consider this quantity in the context of our asymptotic approximations ofthe Pareto-Lorenz potentials, stated in Equations (C.2)-(C.4). Applying Lemma C.1, ζ (cid:20) NW (cid:18) B − w A (cid:19) + (1 − L ) w (cid:21) ≈ ζ (cid:20) N B ∞ W + w (1 − L ∞ ) + 2 CW w f (cid:48) ( w ) e − f ( w ) (cid:21) ≈ ζ (cid:20) N B ∞ W + (1 − L ∞ ) w (cid:21) , where we have used that e f ( w ) (cid:29) w f (cid:48) ( w ) , which was proven in Lemma B.2.We are now equipped to use Lemmas C.1-C.3 to prove that f ( w ) = − log[ P ( w )]reduces to the functional form provided in Equation (2.15). Proposition
C.4.
At large wealth, f ( w ) ≈ B ∞ (cid:90) w dx χ ( x ) x + ζ (1 − L ∞ )2 B ∞ w + 2 ζB ∞ N − T WB ∞ N W w.
Proof.