A simple dynamical model leading to Pareto wealth distribution and stability
aa r X i v : . [ q -f i n . E C ] O c t A SIMPLE DYNAMICAL MODEL LEADING TO PARETOWEALTH DISTRIBUTION AND STABILITY
RICARDO P´EREZ MARCO
Abstract.
We propose a simple dynamical model of wealth evolution. The invari-ant distributions are of Pareto type and are dynamically stable as conjectured byPareto. Introduction.
At the end of the XIXth century, in his studies of wealth and income distributionon different countries, Vilfredo Pareto ([7], [8]) discovered the universal power law thatgoverns the upper tail of wealth distribution. It is well known that this is not a goodmodel for the lower part of the curve that is more dependent on specific sociologicalfactors and of log-normal type (see the discusion in [6]). The exponent in the powerdecay is country dependent and is an indicator of equitative wealth (re)distribution.A larger exponent indicates a more equitative wealth distribution. Pareto’s universalassymptotic behaviour appears in distributions from various other contexts, and, aswe show, is typical from competitive system where the reward is proportional to theaccumulated wealth. The purpose of this article is to provide a simple explanation toPareto’s empirical observation. We propose a natural dynamical model of evolutionof wealth where Pareto distributions emerge as invariant dynamically stable distri-butions of this Dynamical System. The stability of the wealth distribution, which isdifferent from the universality property, was conjectured by Pareto, whose intuitionapparently comes from his empirical observations. We can read in [8], chap. VII,point 31, p.393: Mathematics Subject Classification.
Primary: 91B55, 91B82, 91A60 .
Key words and phrases.
Pareto, distribution, wealth. Wealth and income are proxies of each other in first approximation for our purposes. “Dynamically stable distribution” in the Dynamical System sense not in the probabilistic sense. Si, par exemple, on enlevait tout leur revenu aux citoyens les plus riches, en sup-primant la queue de la figure des revenus, celle-ci ne conserverait pas cette forme,mais tˆot ou tard elle se r´etablirait suivant une forme semblable `a la premi`ere. There are other classical models and studies of Pareto empirical observation andpower laws (like Zipf’s law). For the record we cite a few classical ones: H. Simon[11], D.G. Champernowne [2], B. Mandelbrot [6],etc Simon model [11] for Zipf’s lawis a “genesis model” of the distribution, i.e. it is a model for its creation. Champer-nowne [2] proposed a general multiplicative stochastic model, and B. Mandelbrot [6]explained Pareto law by the universal limit character of Pareto-L´evy probabilisticallystable distributions. 2.
The dynamical model.
In this first section, we propose and study a dynamical model of wealth evolutionwhich is a simple first approximation. . Setup.
Let f ( x ) be the wealth distribution, i.e. df = f ( x ) dx is the numberof individuals with wealth in the infinitesimal interval [ x, x + dx [. The distributionfunction f : R + → R + is continuous, positive and decreasing and lim x → + ∞ f ( x ) = 0.A distribution is of Pareto type if it presents a power law decay x − α at + ∞ , that islim x → + ∞ − log f ( x )log x = α > . The exponent α >
Pareto exponent . A distribution of the form f ( x ) = C.x − α iscalled a Pareto distribution. Smaller values of α indicate larger inequalities in wealthdistribution. Notice that α > ∞ , i.e. finite wealth at infinite (finitness near 0 is not significant since the modelaims to explain the tail behaviour at + ∞ ). . Wealth dynamics.
We focuss on the evolution of individual wealth. We assumethat the evolution is based on two main factors: Finantial decisions, that we modelas a betting game, and by public redistribution of wealth, that absorbs part of theindividual wealth into public wealth.For the first factor we model the finantial decisions of each individual by a sequenceof bets. Each financial decision turns out to be a bet, waging a proportion of hiswealth. As a first approximation, we assume that the probability of success is thesame for all agents and bets 0 < p < “If, for instance, we confiscate all income to the richests citizens, thus erasing the tail of incomedistribution, this shape will not persist and sooner or later it will evolve to a similar shape of theoriginal.” SIMPLE DYNAMICAL MODEL LEADING TO PARETO WEALTH DISTRIBUTION AND STABILITY3 round, each agent risks the same percentage of his wealth, a fraction γ > γ and if helooses his wealth is divided by 1 + γ .Only considering this first factor, one round evolution the distribution transformsinto the new distribution W ( f )( x ) = p γ f ( x/ (1 + γ )) + (1 − p )(1 + γ ) f ((1 + γ ) x ) . The operator W is “wealth preserving”. In terms of L -norm we have ||W ( f ) || L = || f || L . The agents will only risk their capital if there is a positive expectation of gain, thuswe should assume that p > / W with a dissipative parameter κ ≥
1, the dissipativecoefficient , W κ ( f )( x ) = 1 κ W ( f )( x ) = pκ (1 + γ ) f ( x/ (1 + γ )) + (1 − p )(1 + γ ) κ f ((1 + γ ) x ) , so that for κ = 1 the operator is wealth preserving. We name the model for κ = 1the “wealth preserving model”. . Invariant distributions.
Distributions invariant by the evolution operator W κ must satisfy the fixed point functional equation W κ ( f ) = f , that is,(1) f ( x ) = pκ (1 + γ ) f ( x/ (1 + γ )) + (1 − p )(1 + γ ) κ f ((1 + γ ) x ) . We solve this equation in the next section. . Solution of the functional equation.
Considering the change of variables F ( x ) = f ( e x ), equation (1) becomes a functional equation for F : R → R (2) a F ( x + λ ) − F ( x ) + b F ( x − λ ) = 0 , where λ = log(1 + γ ) > a = (1 − p )(1 + γ ) /κ > b = p/κ/ (1 + γ ) > F thatsatisfy a functional equation of the form ω ⋆ F = 0 , R. P´EREZ MARCO where ω is a compactly supported distribution. In our case, ω = a δ λ − δ + b δ − λ .In our simplified model we don’t need the general theory and the equation can besolved by elementary means. First, the exponential solutions are easy to calculate. Afunction F ( x ) = e ρx is a solution if e ρλ satisfies the following second degree equation:(3) a (cid:0) e ρλ (cid:1) − (cid:0) e ρλ (cid:1) + b = 0 . Observe that the discriminant ∆ = 1 − ab is positive since we have ab = p ( p − κ < κ , thus ∆ > − κ > κ ≥ e ρλ = 12 a ± a √ − ab . Since a > P ( x ) = ax − x + b satisfies P (0) > P (1) < x and x with 0 < x < < x . Therefore, we have two familiesof solutions for ρ in two vertical lines in the complex domain, for k ∈ Z , j = 0 , ρ j,k = λ − log x j + 2 πikλ − . Note that ℜ ρ ,k < < ℜ ρ ,k . Let ρ = ρ , < ρ = ρ , >
0. Observe thatthe particular solution F ( x ) = C.e ρ x leads to the solution f ( x ) = F (log x ) = C.x ρ which is exactly Pareto distribution with Pareto exponent α = − ρ .We can now solve the functional equation completely : Theorem 1.
The general solution of the functional equation (2), (4) a F ( x + λ ) − F ( x ) + b F ( x − λ ) = 0 , (with a, b, λ as above) is F ( x ) = e ρ x L ( x/λ ) + e ρ x L ( x/λ ) where L and L are Z -periodic functions. The way to study these equations is by Fourier transforming it (`a la Carleman [1] using hyper-functions in order to work in sufficient generality). One of the general results by L. Schwartz (see[10] Theorem 10 p.894) is the “spectral synthesis” of solutions: Smooth solutions are uniform limitson compact set of R of linear combinations of exponential solutions ( e ρx ) ρ . Also these exponentialsolutions are not limits of linear combinations of the others, thus the expansion is unique. We have a strong form of Schwartz spectral theorem.
SIMPLE DYNAMICAL MODEL LEADING TO PARETO WEALTH DISTRIBUTION AND STABILITY5
In order to solve the functional equation (2), we consider H ( x ) = F ( x + λ ) − e ρ λ F ( x ). Substracting (2) from (3) multiplied by e − ρ λ F ( x ) we get aH ( x ) − be − ρ λ H ( x − λ ) = 0 , or H ( x ) = (cid:18) ba e − ρ λ (cid:19) H ( x − λ ) . Considering ˆ H ( x ) = (cid:18) ba e − ρ λ (cid:19) − x/λ H ( x ) , we have that ˆ H ( x ) = ˆ H ( x − λ ), i.e. there is a Z -periodic function L such that H ( x ) = (cid:18) ba e − ρ λ (cid:19) x/λ L ( x/λ ) . Therefore we have F ( x + λ ) − e ρ λ F ( x ) = (cid:18) ba e − ρ λ (cid:19) x/λ L ( x/λ ) . Now, put ˆ F ( x ) = e − ρ x F ( x ) . Then we need to solveˆ F ( x + λ ) − ˆ F ( x ) = e − ρ λ (cid:18) ba (cid:19) x/λ e − ρ x L ( x/λ ) , if we write G ( x ) = e ρ λ ˆ F ( x ) and c = − ρ + λ − log( b/a ), G ( x + λ ) − G ( x ) = e cx L ( x/λ ) . We use the following lemma:
Lemma 2.
For c ∈ R , λ > , and L a Z -periodic function, the solutions of thefunctional equation (5) G ( x + λ ) − G ( x ) = e cx L ( x/λ ) , are of the form G ( x ) = G ( x ) + M ( x/λ ) , where M is a Z -periodic function, and for c = 0 , G ( x ) = e cx e cλ − L ( x/λ ) , and for c = 0 G ( x ) = λ − x L ( x/λ ) . R. P´EREZ MARCO
Proof.
Obviously in both cases G is a particular solution. Then the functional equa-tion is equivalent to M ( x + 1) − M ( x ) = 0, where M ( x ) = G ( λx ) − G ( λx ), i.e. M is Z -periodic. (cid:3) So, in the non-degenerate case ( c = 0), absorbing the multiplicative constants into L and M , the general solutions of (2) are of the form F ( x ) = e ( − ρ + λ − log( b/a )) x L ( x/λ ) + e ρ x M ( x/λ ) . And coming back to the second degree equation (3) we have e − ρ λ ba = e ρ λ , so F ( x ) = e ρ x L ( x/λ ) + e ρ x M ( x/λ ) . Indeed the degenerate case never happens:
Lemma 3.
We have c = 0 .Proof. If c = 0 then e ρ λ = b/a = e ρ λ .e ρ λ and e ρ λ = e ρ λ , the root of the seconddegree equation would be double and the discriminant would be ∆ = 0 but we haveseen that ∆ > (cid:3) If we request that
F > F ( x ) → x → + ∞ (the only sound solutions)then L = 0 and M > F ( x ) = e ρ x M ( x/λ ) . Finally we have f ( x ) = x ρ M ( λ − log x ) . If we look for continuous solutions, then M must be continuous and bounded since itis Z -periodic, thus f satisfies Pareto assymptoticslim x → + ∞ − log f ( x )log x = − ρ = α > . . Pareto exponent.
It is interesting that we can compute an explicit expressionof the Pareto exponent in terms of the parameters κ , γ and p , Corollary 4.
The Pareto exponent is given by α = − ρ = − λ − log (cid:18) − √ − ab a (cid:19) or α = 1 − log (cid:18) κ − √ κ − p (1 − p )2(1 − p ) (cid:19) log(1 + γ ) . SIMPLE DYNAMICAL MODEL LEADING TO PARETO WEALTH DISTRIBUTION AND STABILITY7
It is interesting to note that the Pareto exponent α decreases when γ increases.This means that a more risky finantial behaviour, or more active economy, favoursunequal distribution. Fortunes are created and lost more often. Ruin is more common.Indeed we know by the Kelly criterion [4] that ruin is almost sure in the long run if γ is larger than a certain threshold. With a slightly modified model we can explainPareto’s theory of “Circulation of Elites”. Indeed this circulation occurs at all level ofsocial status when the agents are not enough conservative to satisfy Kelly criterion.We will discuss these questions in a companion article [9].The Pareto exponent also increases with κ since dαdκ = 1log(1 + γ ) κ − p κ − p (1 − p ) p κ − p (1 − p ) (cid:16) κ − p κ − p (1 − p ) (cid:17) , is positive. This is natural since a larger κ means a larger demographic and fiscalpressure and thus we expect a better redistribution of wealth and a larger Paretoexponent. . A remarkable solution in the wealth preserving model.
A Pareto exponent α > κ = 1, the Pareto exponent is exactly α = 1. Theorem 5.
In the wealth preserving model, κ = 1 , the Pareto exponent is exactlyequal to α = 1 .Proof. For κ = 1 we have κ − p (1 − p ) = (2 p − . Therefore κ − p κ − p (1 − p ) = 2(1 − p ) . And the formula in the previous section gives α = 1. (cid:3) This result is natural and to be expected: For κ < ∞ , and for κ > ∞ . From the form of the invariant solutions, wehave: Theorem 6.
For an invariant solution, the following conditions are equivalent: (1)
The tail wealth is summable, W ( f, x ) < + ∞ . (2) The Pareto exponent α is larger than , α > . (3) The model is wealth dissipative, that is κ > κ . R. P´EREZ MARCO
It has been observed that the Pareto exponent of the wealthiest fraction of thepopulation has a Pareto exponent which is much closer to 1 than expected (or to therest of the medium class, whatever this means). So for this class of the population thedissipative coefficient is closer to the critical one κ , this means that the wealthiestpart of the population is able to avoid the mechanisms of fiscal redistribution ofwealth. . Stability of invariant solutions.
We now study the Pareto problem of stabilityof the Pareto distribution.Since κ >
1, we can observe that for the L -norm the operator W κ is contracting: Lemma 7.
Let f, g : R ∗ + → R + be measurable functions , with f − g ∈ L ( R ∗ + ) , then ||W κ ( f ) − W κ ( g ) || L ≤ κ − || f − g || L . Proof.
We have |W κ ( f )( x ) − W κ ( g )( x ) | ≤ pκ (1 + γ ) | f ( x/ (1 + γ )) − g ( x/ (1 + γ )) | + (1 − p )(1 + γ ) κ | f ( x (1 + γ )) − g ( x (1 + γ )) | and the result follows integrating over R ∗ + . (cid:3) Obviously this lemma is only interesting when || f − g || L is finite. For each invariantsolution f it is natural to consider the space of measurable bounded perturbationsof f for the L -norm, M ( R ∗ + , R ) denotes the space of Borel measurable functions, S f = { g ∈ M ( R ∗ + , R ); || g − f || L < + ∞} . Then the fixed point f is a global attractor in S f and we have: Theorem 8.
For any g ∈ S f , we have that W nκ ( g ) → f for the L -norm at ageometric rate. This proves the Pareto stability conjecture, exactly as stated by Pareto (see thecitation in the introduction): If we remove all wealth larger than some value x fromthe invariant solution, then the perturbation thus obtained is L bounded because ofsummability of the tail, hence the stability.3. Other more refined models.
With the same ideas, we can build more sophisticated models that will be studiedin the future. The main difference with the model presented here is that the invariantsolutions cannot be computed explicitely in general, nor we can give close formulas
SIMPLE DYNAMICAL MODEL LEADING TO PARETO WEALTH DISTRIBUTION AND STABILITY9 for the Pareto exponents. But this does not prevent numerical studies of the invariantsolutions.We may more realistically assume that there are different sorts of individuals withdifferent skills for finantial investment (different p ’s), and different risk profiles (dif-ferent γ ’s). If we assume that each class of individuals is equally represented accrosswealth classes (which is not true, the more skilled ones should be more numerous inthe upper classes), then we end with a general wealth operator of the form W κ ( f ) = X i p i (1 + γ i ) κ f ( x/ (1 + γ i )) + q i (1 + γ i ) κ f ( x (1 + γ i )) , with X i p i + X i q i = 1 . The exponentials of the Pareto exponents appear then as roots of a Dirichlet polyno-mial. One can prove, using results from [10] that the invariant solutions obey Paretolaw.A more realistic model consists in allowing the dissipative coefficient κ to be nonconstant and make it dependent on x . In principle, x κ ( x ) should be increas-ing. Then the search for invariant solutions leads to a functional equation withnon-constant coefficients whose possible explicit resolution depends on the form ofthe function x κ ( x ). Acknowledgements.
I thank my colleague Philippe Marchal for pointing out anerror in the formula of the first version of this article.
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