ANALYSIS OF A MODEL FOR WEALTH REDISTRIBUTION

This paper provides a detailed analytical investigation of a kinetic model for the redistribution of wealth in a simple economy, which was proposed by Chakrabarti, Chatterjee and Manna. Estimates on the rate of convergence towards the steady state are derived, both for the finite-agent model, found in Monte-Carlo simulations, and the continuous model, corresponding to a homogeneous Boltzmann equation. We find that arbitrary Pareto tails with index ν > 1 appear as stationary wealth distribution for a suitable choice of the model parameters. These steady states are always approached in the long-time limit at a sub-exponential rate.

1. Introduction 1.1. Motivation: econophysical models. In the past decade, various kinetic models have been proposed in order to study the redistribution of wealth in a (over-simplistic) market economy. The paradigma behind these Boltzmann-like descriptions is that the laws of statistical mechanics for particle systems also govern -to a certain extend -the trade interactions between agents. The founding idea is a tight analogy between trade events of agents on a market and collisions of particles in a gas. Just as a classical kinetic model is defined by prescribing the collision kernel for the microscopic particle interactions, the econophysical model is defined by prescribing the exchange rules for money in trades. Up to now, there is little a priori knowledge about which factors should enter into the exchange rules (in order to make the model realistic), and which should not (in oder to keep the final equations as simple as possible). The value of a proposed model is estimated a posteriori by comparison with real-world data. The most widely accepted benchmark is the shape of its steady money distribution: realistic models should exhibit a rich "upper class" of agents. More precisely, if P ∞ (w) denotes the density 1 of agents with wealth w > 0 in the steady state, then P ∞ (w) should exhibit a Pareto tail, The exponent ν is referred to as Pareto index, named after the economist Vilfredo Pareto [22], who proposed a law of the form (1) at the end of the 19th century, see also [18]. The Pareto law (1) applies in various economic contexts, for instance, to the total wealth distribution among the population in a western country. According to the existing data from the past decades, the index ν usually ranges between 1.5 and 2.5. The features typically incorporated in kinetic trade models are saving effects and randomness. Randomness means that the amount of money changing hands is non-deterministic. Saving means that each agent is guaranteed to retain at least a certain minimal fraction of his initial wealth at the end of the trade. A third -and very decisive -ingredient for the vast majority of models is the pointwise conservation of money in individual trades; this corresponds to the strict conservation of energy in collisions in ideal gases. Studies on this particular class of models constitute a rapidly growing field at the borderline between economics and statistical physics, counting dozens of publications within the last few years; see the book [12] for an overview on recent research. Numerous numerical simulations for models of the prescribed type have been carried out with different mechanisms for saving and varying degree of randomness. One observation is that the Date: October 30, 2007. 1 Here we adopt the unfortunate choice to denote by P the density, not the distribution function; this abuse of notation is common in the econophysics literature. particular realization of randomness has little influence on the shape of P ∞ , and can usually be substituted by a deterministic process [9,10]. Another observation is a tendency to fall into one of two extremal cases: For part of the models, P ∞ has an exponentially small tail [1,5,6,15]; for the other part, Pareto tails were found [10,11,17], but were of critical index ν = 1, corresponding to a steady distribution of infinite total wealth. So far, very few simulations have produced Pareto tails with a realistic index. Here, we restrict attention to the model with quenched saving propensity, introduced by Chatterjee, Chakrabarti and Manna [11], called CCM-model in the following: each agent carries a personal spending propensity 2 -a parameter s ∈ (0, 1 2 ) -which varies from agent to agent and does not change with time. In a binary trade between agents i and j, money changes hands according to the rule Here w i , w j and w * i , w * j denote the agents' wealth before and after the trade, respectively, and s i , s j are the respective personal spending propensities. This model is clearly pointwise conservative, w * i + w * j = w i + w j . The quantity ∈ [0, 1] is a random variable, but we limit ourselves to the deterministic case ≡ 1 2 . The defining parameter for the CCM-model is the probability density ρ on (0, 1 2 ), according to which the spending propensities are distributed among the agents. First numerical experiments produced either an exponentially small tail or a Pareto tail of critical index. It has then been argued by Mohanty [20] that arbitrary tails can be created if the density ρ is suitably chosen: a steady state for (2) is attained if there exists a constant M s.t. s i w i = M for all agents, corresponding to the wealth distribution The tail of this distribution is completely determined by the behavior of ρ(s) in an arbitrarily small neighborhood of s = 0. Here, we establish conditions on ρ under which P * in (3) appears as the long-time limit of solutions to the underlying kinetic model, and we estimate the rates of convergence towards the corresponding steady state. What we find is that the rate of convergence is related to the fatness of the tail: If P * has an exponentially small tail (or is compactly supported), then the steady money distribution is attained exponentially fast. If P * is of Pareto type with Pareto index ν > 1, then relaxation happens on an algebraic time scale only, with rate t −(ν−1) . The convergence behavior we find is in agreement with recent numerical experiments and semianalytic considerations, see e.g. [8] and references therein. For completeness, we remark that there are numerous theoretical studies in order to describe the kinetic dynamics, the shape of the steady state and the long-time convergence for the model at hand, e.g. [10,14,17]. However, these works usually focus on appropriate limits or simplifications of the Boltzmann equation. On the other hand, rigorous results are available for a variety of related "non-standard" Boltzmann models of Maxwell type; the latter have attracted a lot of attention in the kinetic community during the last decade. We refer to [3] and references therein for first rigorous investigations on the relaxation behavior of inelastic Maxwell molecules, and to [4] for some more recent results and generalizations. The specific equations considered here, however, do not fit into the framework of the mentioned papers. Thus, the results presented below are (to our knowledge) the first rigorous ones for the full kinetic equation resulting from the CCM model. Our investigation is performed in two ways. First, we analyse the steady states of the kinetic model with finitely many agents, as it appears in the context of Monte Carlo simulations. Second, we investigate the Boltzmann equation associated to the kinetic model with an infinite number of agents. The results of these sections are briefly summarized below. We close with a remark concerning the modelling. Even though we prove convergence towards P * from (3) in almost all cases, this should not be understood as an appreciation of the CCM-model.
On the contrary, our results rather indicate that the appearance of Pareto tails is unnatural and fragile in this framework. Not only is the complete structure of the tail of P * destroyed by an arbitrarily small variation of the model parameter ρ(s) near s = 0. Also, the slow algebraic convergence rate towards the steady money distribution means that the steady state will be dynamically unstable under perturbative effects. In comparison, Pareto tails of adjustable index appear naturally once one breaks with the paradigma of pointwise conservation of money. In several models for open economies, where wealth increases due to trade interactions, the money distribution curve attains a self-similar profile with a Pareto tail, see e.g. [13,21,23]. Likewise, a combination of trades with risky investments, which preserve money in the statistical mean, but not in individual trades, are able to generate Pareto behavior [19]. In all mentioned models, the Pareto index depends in a robust way on the parameters for the microscopic interaction, and the steady state is exponentially attracting.
1.2. Analysis of Monte Carlo simulations. In section 2, we analyse the kinetic model with exchange rules (2) for finitely many agents. This is the typical situation encountered in Monte-Carlo simulations, which have been described in detail e.g. in [7]. A particular realization of the finite-agent model is defined by the number N > 1 of participating agents and the individual spending propensities s i ∈ (0, 1 2 ), which are sampled in accordance to the given probability density ρ. The state of the system at a particular time corresponds to the N numbers w i ≥ 0, representing the current wealth of the agents. We prove that each realization possesses a unique steady state, and that the w i converge towards their respective limits W i = M/s i as the number of successive trade interactions goes to infinity.
The key tool is the use of the entropy functional H defined in (9), which monotonically decreases with the number of trades. Subsequently, we estimate the speed of convergence by relating the entropy production, i.e. the average change of H by any particular trade, to H itself. We find that the convergence is exponential in the number of trades, but with a rate inverse proportional to the smallest appearing saving propensity s i > 0. Thus, the relaxation time strongly depends on the specific realization. Next, we introduce the configurational average P N : R + → R + , which is the average over the stationary wealth distributions for all possible realizations of the N -agent model with fixed parameter ρ. The average of the results from a sufficiently large number of Monte-Carlo simulations is an approximation to P N . For the limit of infinitely many agents, N → ∞ (keeping the average wealth m > 0 per agent fixed) we prove that if then P N converges uniformly to the steady state P * given in (3). If (4) is violated, then P N converges (weakly in the sense of measures) to a Dirac distribution at w = 0. We remark that (4) is violated for a uniform measure ρ ≡ 2 · 1 (0,1/2) , which is frequently used in the CCM-model. Finally, we calculate the tail shape of P N explicitly for finite N , when ρ(s) coincides with a power function in an arbitrarily small neighborhood of s = 0. P N displays the finite-size correction of the Pareto tail of P * , which forms in the limit N → ∞. In particular, we obtain a formula for the finite-size Pareto tail of critical index one, which has been the object of extensive studies, [7,10,17].
1.3. The Boltzmann equation. To free the model from finite-size effects, we pass to the related homogeneous Boltzmann equation (22) in section 3. The latter is the evolution equation for the density f (t; w, s) of agents with wealth w ≥ 0 and spending propensity s ∈ (0, 1 2 ) at time t ≥ 0. The collision kernel in (22) is defined through the rules (2). The corresponding wealth distribution P (t; w) is given by the w-marginal of f , and the model parameter ρ constitutes the time-independent s-marginal. A consequence of our results is that the marginal P (t) converges weakly (with explicit rates) to P * given in (3), under mild hypotheses on the initial condition and provided that (4) holds. If (4) is violated, then P (t) converges weakly to a Dirac distribution at w = 0.
Probably more interesting than the convergence of P (t) is the relaxation behavior of the solution to the underlying kinetic equation. Define the mean wealth W (t; s) of agents with a certain spending propensity s ∈ (0, 1 2 ), This quantity satisfies the evolution equation The unique steady state of the Boltzmann equation (22) is the density f ∞ which is concentrated on the graph where W ∞ is a stationary solution to (5). More precisely, if (4) holds, then W ∞ (s) = M s −1 , where M > 0 is related to the average wealth. If (4) is violated, then W ∞ ≡ 0. The main result, formulated in Theorem 8, is that under mild conditions on the initial data and the parameters, f (t) concentrates on the graph which itself converges to W ∞ . Moreover, in the regime corresponding to Pareto tails of index 1 < ν < 2 for P * , one finds a clear separation of time scales: In the first stage, f (t) concentrates on M(t). In the second stage, the concentrated measure moves with the graph M(t) towards the limit f ∞ . For a Pareto index ν close to one, the two time scales are approximately separated by a factor t (1−ν)/2 . Correspondingly, the proof consists of two steps. First, the speed of convergence of W (t) towards its limit W ∞ is estimated, using suitable entropy functionals that decay along solutions of (5). As expected, the convergence rate is directly connected to the vanishing behavior of ρ(s) near s = 0; for an exponentially small tail of P * , one finds exponential convergence, for a Pareto tail, the rate is algebraic. Second, an evolution equation for the w-variance of f (t) with respect to the mean W (t) is derived. The solution of the latter equation can be estimated from the results of the first step. We stress that, despite its simple appearance, equation (5) is not completely trivial to analyse. As a by-product of our results, we obtain an asymptotic expansion of its solutions in Proposition 1 in terms of s −1 , with rigorous error estimates. In fact, our results for (5) carry over -word by word -to the following generalization of the radiative transfer equation [16], in which µ is a given probability measure on the domain Ω, and s : Ω → R + is some prescribed function. The speed of convergence of W (t) to the steady state is then connected to the vanishing behavior of s.

1.4.
Notations. For us, the set R + = [0, +∞) includes zero. By abuse of notation, we shall frequently abbreviate 1 2 0 φ(s) · ρ(s) ds by φ(s) dρ(s). Even worse, we shall occasionally write φ(w, s) · f (w, s) dw ds in cases where f is not a function but merely symbolizes the density of a measure f (w, s) dw ds on R + × (0, 1 2 ), possibly with concentrations. Finally, in proofs, C denotes a generic (positive and finite) constant, which may vary from one line to the next.

Analysis of Monte Carlo simulations
2.1. Convergence to the steady state. There is a fixed number N of agents participating in the simulated game. Denote by s i the spending propensitiy of the ith agent, with 0 < s i < 1 2 , so that λ i = 1 − 2s i is the saving propensity in the original CCM-model [11]. For simplicity of notation, assume that the parameters s i are ordered, s 1 ≤ s 2 ≤ · · · ≤ s N . The state of the system at an instant of time is completely described by N non-negative real numbers w i , which denote the respective wealth of the ith agent. Some initial distribution of the money is given, i.e. there are N numbers w i ≥ 0 prescribed. We assume that the average wealth per agent m > 0 is a fixed parameter, so that N · m = N k=1 w k . A time step consists of the following: • Two distinct indices i = j between one and N are randomly picked.
• The numbers w i and w j change according to the collision rules (2). In other words, replace w i and w j , respectively, by • All other w k are left unchanged.

Lemma 1. The wealth distribution with
constitutes a steady state for the dynamics.
Proof. Simply consider rules (6)&(7) for two arbitrary agents i and j. One finds that The normalization constant M is chosen so that the mean wealth is m.
In order to study the convergence to equilibrium, introduce the deviations To measure the distance of a given wealth distribution from the associated steady state, define This quantity constitutes a Lyapunov functional for the dynamics.
where i and j are the indices of the agents which have interacted in trading, and the quantities ∆ i and ∆ j denote the values before the trade took place.
Proof. We only need to investigate the changes of s i ∆ 2 i and s j ∆ 2 j due to the trade. By linearity of (6)&(7), and since the W k constitute a steady state, the differences ∆ * i and ∆ * j after the trade satisfy All the other components ∆ k remain unaltered. This gives (10).
By (10), the "distance" H cannot increase. Moreover, since s k < 1 2 by assumption, H is strictly reduced by the trade unless s i ∆ i = s j ∆ j . Recall that ∆ k = 0; hence s 1 ∆ 1 = · · · = s N ∆ N implies that ∆ 1 = · · · = ∆ N = 0. In conclusion, as long as the system is not in equilibrium, there is always a pair of agents whose "collision" on the market will strictly reduce H. (8) is the unique one. Moreover, as the number of collisions goes to infinity, each w i converges to the respective W i , for i = 1, . . . , N , almost surely. This is independent of the initial wealth distribution.

2.2.
Rate of convergence. Below, we provide an estimate on the rate at which the wealth distribution approaches its steady state. The key ingredient are the estimate (10) above and the following where s i > 0 are the spending propensities.
Proof. Since the sum of the ∆ i is zero, we obtain where the estimate follows from Hölder's inequality for sums. Consequently, and the claim (11) follows since s 1 is the smallest of the s k .
Denote by H * the expectation value of H after one trade between two randomly choosen agents i and j. According to (10), Estimate (11) now yields the relation Iterations lead to an estimate for H T , the value of H after T consecutive trades: Define t = T /N as the time variable of the simulations, so that the unit time interval corresponds to one trade per agent on the average. By definition of H, one obtains exponential relaxation of w in L 2 in the statistical mean, Although (13) constitutes only a lower bound on the speed of convergence, numerical experiments indicate [8] that the temporal rate in this estimate is sharp. The size of s 1 > 0 -which depends on the specific realization of the experiment -is seemingly decisive for the relaxation process. This is disturbing in particular since the corresponding average relaxation time (s 1 ) −1 depends on the system size N in general, or may even be infinite.

2.3.
Shape of the tails. This section provides a calculation of the shape of the ensemble-averaged density P . Let us recall how P (w) is calculated from the numerical simulations [7]: a series of K > 1 experiments is carried out, in which the number of participating agents N and the average wealth per agent m is fixed. The spending propensities s i ∈ (0, 1 2 ) are assigned to agents according to a fixed law with continuous density ρ(s) at the beginning of each experiment. By the previous considerations, there is a unique steady state for the discrete game, given by (8), and it is approached exponentially fast. To the steady state of the kth game, the empirical distribution is associated, δ wj (w) being the Dirac distribution centered at w = w j . Finally, the ensemble density P N is calculated as the average of these single distributions, We propose the following definition of P N : Definition 1. Let N independent, identically distributed (according to ρ(s)ds) random variables s i on (0, 1 2 ) be given. To these, associate the normalization constant Then P N is the probability density of w 1 = M N /s 1 , the wealth of the first agent in the steady state.
Notice that, since we do not assume ordering of the s i in this definition, the first agent is not distiguished from the others. Our main interest is the convergence of the P N towards the limit distribution as introduced by Mohanty [20], where Γ is the expectation value of s −1 1 , But the latter might be infinite, so that (14) is not well-defined. We obtain the following: Theorem 2. Let the density function ρ : (0, 1 2 ) → R + be given. • If ρ is continuous, with lim s→1/2 ρ(s) = 0, and such that Γ < +∞, then each P N (w) is a continuous function on R + with support in [0, N · m], and P N (0) = 0. Moreover, P N converges to P ∞ as N → ∞ in the sense that • If Γ = +∞, then P N converges weakly towards a Delta distribution at w = 0 as N → ∞.
The convergence in (16) guarantees a slightly better approximation of the tail of P * then just uniform convergence on R + .
Proof. In order to derive a formula for P N , define yet another random variable which has some density η N on R + , which can in principle be calculated from ρ. We arrive at an alternative representation of w 1 , Elementary calculations give Further, by independence of s 1 and S N , In summary, with the convention that ρ(s) = 0 for s outside the interval [0, 1 2 ]. In the representation (18), we have used that s i ≤ 1 2 and hence S N ≥ 2(N − 1)/N ≥ 1 to improve the lower limit of the integral. Now, assume that ρ is continuous, and Γ < ∞. Continuity of P N follows immediately by the representation (18). Furthermore, using that ρ(s) = 0 for s ≥ 1/2, where R := sup ρ < ∞ by continuity of ρ. The last expression converges to zero as w 0, since η N is the probability density of S N , which has finite expectation value by assumption. This proves that P N (w) → 0 as w → 0. Let some > 0 be given. By the law of large numbers, P |S N − Γ| < δ → 0 as N → ∞ for any fixed δ > 0. For each δ > 0, there exists an N 0 ∈ N such that If w ≤ m/Γ and δ < Γ/2, then it is easily checked that both terms inside the supremum are zero.
On the other hand, it follows from continuity of ρ that this supremum can be made smaller that R -uniformly in w ≥ m/Γ -by diminishing δ and increasing N 0 accordingly. In conclusion, for all w > 0, In the case that the integral in (15) diverges, it is another application of the law of large numbers that for each fixed γ > 0, one has P S N < γ → 0 as N → ∞. Consequently, for any W > 0, can be made smaller than any given > 0 by choosing δ > 0 suitably, and having N large enough. This means that all mass concentrates at w = 0 as N → ∞.
If ρ(s) behaves like a power function near s = 0, then the tail of P N is explicit.
Theorem 3. Assume that ρ(s) = a · s α with some a > 0, α ≥ 0 for 0 ≤ s ≤s. Then there exist constants G ρ N > 0 such that the corresponding density functions P N satisfy We emphasize that the hypothesis on ρ(s) is only concerned with its behavior near s = 0, and the conclusions hold for P N on the whole range of large w. In order to estimate the magnitude of the tail, i.e. the values of G ρ N , one needs additional information on ρ.
Proof. Clearly, P N (w) = 0 for w > N · m by Definition 1. Assume that w ≥ m 2s ; one easily verifies that the argument of ρ in (18) is always less thans on the whole range on integration. Hence, by the prescribed homogeneity of ρ, This proves the claim (20) on the shape of the tail of P N . For α > 0, convergence of the G ρ N follows from Theorem 2. It remains to prove the decay of G ρ N if α = 0. Recall that ∞ 0 w dP (w) = m, the average wealth m per agent, which is independent of N . But by (20), which implies G ρ N ≤ m/ log(2Ns). The proof in the case ρ(s) = 0 is an obvious alternation of the argument above.

Analysis of the Kinetic Model
3.1. The Boltzmann equation. It is one paradigma of econophysics that the evolution of a system of very many interacting agents is modelled by the homogeneous Boltzmann equation. In the model at hand, the Boltzmann equation describes particles which have two characteristics: their (time-dependent) wealth w ≥ 0 and their (fixed) spending propensity s ∈ (0, 1 2 ). Denote by f (t; w, s) the probability density of the agents possessing wealth w and spending propensity s at time time t > 0. Then f satisfies where the collisional gain operator Q + describes the redistribution of wealth due to the binary trade interactions of agents. With the collision rules defined in (6)&(7), the weak form of equation (21) reads Here Φ : R + × (0, 1 2 ) → R denotes an arbitrary -non-negative and sufficiently regular -test function. It is a classical result, see e.g. [2], that for a prescribed initial probability density there exists a measure-valued solution f (t) satisfying (22). We are going to study the convergence of f (t) for large times. Assume that the s-marginal of f 0 coincides with ρ, the prescribed density of the saving propensities, By substituting a w-independent test function Φ(w, s) = φ(s) into (22), one verifies that the respective integral is time-independent, Hence the s-marginal of f (t; w, s) is always ρ(s), in accordance to the fact that agents do not change their spending propensity in time. Next, introduce the local mean wealth W (s) for s in the support of ρ by Using test functions of the form Φ(w, s) = wφ(s) in (22), one obtains This is a weak form of The following section 3.2 is devoted to the study of (23). We defer the further analysis of (22) to section 3.3, where we prove that f (t; w, s) concentrates on the graph defined by w = W (t; s).

Convergence properties of W (t). The equation (23) is linear,
and the operator A is well-defined and bounded on L 1 ρ . To any given non-negative initial datum W 0 ∈ L 1 ρ , there exists a unique solution W (t); this solution is non-negative and preserves the mean wealth, m := W 0 (s) dρ(s) = W (t; s) dρ(s).
For the following asymptotic analysis, assume the following bound on the initial data W 0 ∈ L 1 ρ , This is the unique positive steady state for (23) with given total mean wealth m > 0. If the integral in (25) diverges, then W (t) converges to W ∞ ≡ 0 pointwise, but not in L 1 ρ . The rates of convergence are now studied in detail, using a hierachy of Lyapunov functionals. Introduce functions J k (s) for k = 0, 1, 2, . . . recursively by Clearly, each J k satisfies the same linear equation as W itself, Further, define the associated entropy functionals Note that F k is the variance of sJ k−1 (s) and thus measures the deviation of W (s) from a steady state.
Proof. The bound (24) implies that W (t; s) ≤ Ks −1 at any positive time t > 0. This follows easily since Ks −1 constitutes a steady state of the equation (23), which imposes a maximum principle on its solutions. Thus G 0 = sW (t; s) 2 dρ(s) ≤ K W (t; s) dρ(s) = K ·m. Furthermore, J 1 (s) ≤ sW ≤ K is bounded, and so are all J k with k ≥ 1. This proves finiteness of F k and G k . Next, observe that dJ k /dt = −A J k = J k+1 by definition. Hence, By definition of J k (s), the integral J k (s) dρ(s) vanishes for k ≥ 1; this yields the first relation in (28). Furthermore, Again, J k (s) dρ(s) vanishes for k ≥ 1, producing the second relation in (28).
In particular, all F k and G k are non-increasing in time, except for F 0 = W (s) 2 dρ(s). In fact, they converge to zero as t → ∞, except for F 0 and possibly G 0 . From the structure of the differential equation hierachy, it is immediate to draw conclusions on their decay behavior. It is natural to distinguish two cases, namely the one where the spending propensity of all agents has a positive lower bound, and the one where the infimum of the spending propensities is zero. In terms of the linear operator A, this corresponds to the existence and the absence of a spectral gap at zero, respectively.

Lemma 5.
Under the assumption (29), the entropies F k with k ≥ 1 decay exponentially fast, Proof. From (28), one has d dt It is not hard to conclude that the given temporal relaxation rates cannot be improved without further restrictions on ρ or the initial condition W 0 .
By Hölder's inequality, the definition of F 1 and (30), . Moreover, by conservation of the mean wealth, In conclusion, proving (31). The pointwise estimate (32) follows from the integral representation of (27) for J 1 , Using (30), it is immediate to estimate the modulus of J 1 ,

No spectral gap.
If the spectrum at zero is closed, i.e. (29) is violated for all > 0, one generically finds algebraic decay of the solution instead of exponential one. In the following, let µ ≥ −1 be an exponent such that Introduce correspondingly for k = 0, 1, 2, . . ., Notice that (33) is always satisfied with µ = −1. However, the quantity Γ in (25) is well-defined iff µ ≥ 0 is possible in (33). Also, recall that by (3) a Pareto tail of index ν > 1 in (1) corresponds to a density satisfying ρ(s) ∝ s ν−1 near s = 0. Correpondingly, one may choose an arbitrary µ > 0 with µ < ν − 1 in (33). Lemma 6. The entropies F k and G k with k ≥ 1 converge monotonically to zero and satisfy with α k from (34) and suitable finite constants C k depending only on m and K in (24).
Proof. First assume that µ ≥ 0 in (33). As in the proof of Lemma 4, we remark that W (s; t) ≤ Ks −1 , and hence |J 1 (s; t)| ≤ K for all t ≥ 0. By Hölder's inequality, and the first integral on the right-hand side is finite by condition (33). In combination with (28) for k = 1, one thus obtains with a positive constant c > 0. Comparision with the solution of the ODEẋ = −cx (2+µ)/(1+µ) yields estimate (35) for F 1 . All remaining inequalities now follow inductively from (28): since G k is positive and monotonically decreasing, one deduces so that G k (t) ≤ F k (t/2)/t, which eventually yields (35). A similar argument holds for F k . Finally, if µ ≤ 0 in (33), one uses that 0 ≤ G 0 (t) ≤ K · m, and that F 1 is monotonically decreasing in order to conclude This provides the estimate (35) for F 1 . The remaning estimates now follow inductively, as above.
As a consequence of these estimates, we obtain an asymptotic formula for W (t).
Proposition 1. Assume (33) with µ ≥ −1. Denote β k = sJ k (s) dρ(s) and define For arbitrary k ≥ 1, and provided that α k > 1, there are finite constants C k , C α such that where α is arbitrary with the restriction 1 ≤ α ≤ α k . If α k = 1, then the the right-hand side in (37) is replaced by Notice that β k (t) = (−1) k+1 ∂ k t W (t; 0), the signed time derivatives of W (t; s) at s = 0. A particular consequence of Proposition 1 is that which is obtained from (37) with k = 1 and α = 1.

Proof. It is easy to check by induction that
Multiply equation (40) by s α , with α ≥ 1. To treat the first term on the resulting right-hand side, observe that s α e −st ≤ C(1 + t) −α for all 0 ≤ s ≤ 1/2 and t ≥ 0. The integral is estimated as follows: The integral converges for T → ∞ if α k > 1. Provided that α ≤ α k , one directly concludes which, in view of (39), proves (37). For α k = 1, one picks up an additional factor log(1 + t) from the integral in (41). Boundedness of the β k for k ≥ 1 is directly concluded by another application of (35).
We are now able to establish a rate for the convergence of W (t) to equilibrium W ∞ is L 1 ρ . In view of the previous lemma, the speed of convergence is essentially determined by the behavior of β 0 (t).
Observe that one deduces the pointwise estimate by combining (43) with (37).
Theorem 6. Assume that the integral in (25) diverges. Then W (t; s) converges pointwise to zero as t → ∞, but does not converge in L 1 ρ . In particular, if holds for all small > 0 with some a > 0 and 0 < η < 1, then with a suitable finite constant B 0 > 0. Finally, if the density ρ(s) is right-continuous at s = 0, with ρ(0) > 0, and initially W 0 (s) ≤ W < ∞, then with suitable constants 0 < b 0 < B 0 < ∞, where b 0 depends on W .
Formula (47) describes the generic situation where ρ is continuous at zero with ρ(0) > 0. For instance, the model with uniformly distributed spending propensity falls into this class. Estimate (47) in combination with (38) indicates that a Pareto tail of critical index ν = 1 seems to form on a time scale t −1 -but collapses to zero on the scale (log t) −1 .

Relation to the radiative transfer equation.
In the preceeding discussion, we performed a detailed asymptotic analysis of solutions to equation (23). Using the infinite hierachy of Lyapunov functionals F k and G k , our analysis resulted in the asymptotic expansions stated in Proposition 1. When one is only interested in L 1 ρ -convergence of W (t) to W ∞ , then there is an alternative, more elegant way to obtain (slightly suboptimal) decay estimates. Equation (23) is formally similar to the radiative transfer equation [16], for u : Ω → R ≥0 on a domain Ω with probability measure m. All entropy functionals of the form with arbitrary convex functions ϕ, decay exponentially in time. The corresponding entropyentropy production estimate reads [16]: This estimate can be applied to (23), leading to Theorem 7. Let a convex function ϕ be given and define where W (t) satisfies (23), and Γ is given in (25).
• If the finite-gap estimate (29) holds with > 0, then • If the no-gap estimate (33) holds with µ > 0, then In fact, since the function ψ : R + → R + with ψ(r) = |r − M | can be approximated in C 0 by smooth convex functions ϕ, the estimates (54) and (55), respectively, directly imply convergence of W (t; s) to W ∞ = M s −1 in L 1 ρ at the given rates. Moreover, employing similar techniques as in the previous section, one can derive pointwise bounds from there, like (44). Notice, however, that the decay in (54) is not optimal in general, see (31).

3.3.
Analysis for the full Boltzmann model. In this terminal section, we establish the connection between the solution W (t; s) of the linear equation (23) and the solution f (t; w, s) of the nonlinear Boltzmann equation (22). Introduce for s in the support of ρ the local variances of wealth, where M 2 denotes the -local -second moment, We shall assume that the integral over the second moments is finite initially, and hence also V (0; s) dρ(s) < +∞.
The quantity V measures the (de)concentration of the density f (t) on the graph .
In fact, if µ t denotes the measure induced by f (t) on R + × (0, 1 2 ), then by Chebyshev's inequality, Our aim is to estimate the time decay of the integral on the right-hand side.
To have an idea of the concentration behavior of f (t), consider the Boltzmann equation (22) in the regime of grazing collisions. The corresponding trade rules read i.e. the spending propensities are scaled by some small parameter ε > 0. Assuming that the collision frequency increases accordingly by a factor of 1/ε, one obtains in the limit ε → 0 for an arbitrary test function Φ, It is immediately seen that the local mean wealth W (t; s) evolves according to (23) as in the original model. Moreover, the second moment M 2 (s) satisfies

For the evolution of the variances
Hence, for each s > 0, the wealth of agents with spending propensity s > 0 concentrates exponentially fast around the mean value W (t; s). In finite-gap situation of (29), one thus obtains exponential concentration at rate exp(− t) in (57). In the no-gap situation (33), one obtains for the integral in (57) Hence concentration in (57) occurs at least at a rate (1 + t) −2 . Now recall that the pointwise convergence of W (t) to W ∞ takes place at the rate (1 + t) −µ , see (44). If this rate is optimal with some 0 < µ < 1, then concentration of f on M(t) happens faster than convergence of M(t) to its limit M ∞ . In fact, one may choose δ(t) = δ 0 · (1 + t) −1 in (57), which shows that -after an intermediate time -an arbitrarily large fraction of the agents is δ(t)-close to M(t), whereas M(t) and M ∞ are apart by approximately δ(t) · (1 + t) 1−µ . In this situation, one finds a clear separation of time scales for the convergence to equilibrium in the simplified Boltzmann equation (59): In the first phase, agents with a certain spending propensity accumulate at the respective mean wealth W (t; s). In the second phase, the wealth distribution closely follows the graph M(t) defined by W (t) to the equilibrium distribution W ∞ from (26). In the following, we show inasmuch this qualitative behavior carries over from the grazing collision limit (59) to the full Boltzmann equation (22). We shall restrict ourselves to the no-gap case (33), leaving the -much easier -finite-gap situation (29) to the reader.
Theorem 8. Assume that ρ satisfies condition (33) with −1 < µ < 1, and that the total variance is initially bounded, cf. (56). Then, with a suitable finite constant C, or C δ , respectively, In particular, f (t) concentrates on the graph M(t) at the given rate.
Proof. Using test functions of the form Φ(w, s) = w 2 · φ(s) in (22), one derives a weak formulation of an evolution equation for M 2 ; the corresponding strong form reads Taking into account that we obtain for the wealth variance In particular, the integrated variance satisfied the equality Since F 1 is integrable in time, and the total variance is finite initially by (56), it follows that the latter remains uniformly bounded for all t ≥ 0. As V is non-negative, it follows further that We continue with detailed calculations for the case µ > 0, and briefly comment on −1 ≤ µ ≤ 0 afterwards. The solution V to (62) can be represented as Multiplication of (64) by s 2 and integration with respect to ρ(s) yields, after trivial manipulations, We estimate the right-hand side term by term: Using (37) with k = 1 and α = 1, since we have assumed µ ≤ 1. Compute the integral from 0 to T of the last expression: And with a similar idea, one shows We turn to estimate the term involving Z itself. For η ∈ (0, 1), we find the recursion relation (see Proposition 1), which holds for an arbitrary δ < 1.
As a consequence of Theorem 8 above in combination with Theorem 5 or Theorem 6, respectively, one can obtain convergence of the wealth distribution function P (t, w) := f (t; w, s) ds towards the limit P * given in (3), or to a Dirac distribution at zero, respectively. The type of (weak) convergence will depend on the regularity of the density ρ. Assuming that ρ is a bounded function, one obtains convergence in distribution, with an explicit rate. We prove the corresponding result in the most relevant situation µ > 0 in (33), leaving the other -slightly easier -cases to the interested reader.
Proof. The function P * (w) coincides with the w-marginal of the density f ∞ (w, s), which is concentrated on the graph and has s-marginal ρ. Hence, withŝ := mΓ −1ŵ−1 , Define µ t as the measure on R + × (0, 1 2 ) associated to f (t). Then, for δ > 0 sufficiently small, We estimate the last sum term by term.
Choosing δ = At −µ with A > 0 large enough yields one half of the inequality (71), namely The other half is proven in a similar way.

Conclusions
The relaxation behavior of a kinetic equation, which constitutes a model for trading on simple markets, has been analysed. We have shown the existence and uniqueness of an attracting steady state, both for the finite-agent model and for the limiting continuous Boltzmann equation. For solutions to the latter, we obtained explicit rates for the two determining processes of relaxation to equilibrium: the first is concentration of the solution on a lower-dimensional subset of phase space, the second is convergence of the latter subset to its equilibrium position. In a certain regime, these two processes happen (in this order) on separated algebraic time scales. The main analytical tool was the use of suitable Lyapunov functionals. In our investigations, we restricted attention to a specific model. However, certain features like slow (algebraic) relaxation in the fat tails seem to be typical for the whole class of pointwise conservative interactions. Properties of this class will be investigated in further work, in particular in comparison with alternative approaches describing open economies or risky markets.