Mean-field theory of an asset exchange model with economic growth and wealth distribution
W. Klein, N. Lubbers, Kang K. L. Liu, T. Khouw, Harvey Gould
MMean-field theory of an asset exchange model with economicgrowth and wealth distribution
W. Klein, ∗ N. Lubbers, † Kang K. L. Liu, ‡ and T. Khouw Department of Physics, Boston University, Boston, Massachusetts 02215
Harvey Gould
Department of Physics, Boston University, Boston, Massachusetts 02215 andDepartment of Physics, Clark University, Worcester, Massachusetts 01610
Abstract
We develop a mean-field theory of the growth, exchange and distribution (GED) model intro-duced by Kang et al. (preceding paper) that accurately describes the phase transition in the limitthat the number of agents N approaches infinity. The GED model is a generalization of the Yard-Sale model in which the additional wealth added by economic growth is nonuniformly distributedto the agents according to their wealth in a way determined by the parameter λ . The model wasshown numerically to have a phase transition at λ = 1 and be characterized by critical exponentsand critical slowing down. Our mean-field treatment of the GED model correctly predicts theexistence of the phase transition, critical slowing down, the values of the critical exponents, andintroduces an energy whose probability satisfies the Boltzmann distribution for λ <
1, implyingthat the system is in thermodynamic equilibrium in the limit that N → ∞ . We show that thevalues of the critical exponents obtained by varying λ for a fixed value of N do not satisfy theusual scaling laws, but do satisfy scaling if a combination of parameters, which we refer to as theGinzburg parameter, is much greater than one and is held constant. We discuss possible implica-tions of our results for understanding economic systems and the subtle nature of the mean-fieldlimit in systems with both additive and multiplicative noise. ∗ [email protected] † Present address: Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, NewMexico 87545 ‡ Present address: Department of Physics, Brandeis University, Waltham, Massachusetts 02453 a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b . INTRODUCTION Agent-based asset exchange models have become useful [1–11] for studying the effects ofchance on the distribution of wealth. These models consist of N agents that can exchangewealth through pairwise encounters. Examples of the exchange mechanism include thetransfer of a fixed amount of wealth and the exchange of a fixed percentage of the averageof the wealth of the two agents. The common feature of these models is that the winner inthe exchange is determined by chance.Of particular interest is the Yard-Sale model [2, 5–8, 12–19] in which pairs of agents arechosen at random and one is designated as the winner with a probability usually taken tobe 1/2. The winner receives a fraction f of the wealth of the poorer agent. The result isthat after many exchanges, one agent gains almost all of the wealth, a phenomena knownas wealth condensation.In this paper we study a generalization of the Yard-Sale model [20] in which a fixedpercentage µ of the total wealth is added to the system after N exchanges. The addedwealth is distributed according to ∆ w i ( t ) = w λi ( t ) (cid:80) Nj =1 w λj ( t ) , (1)where w i ( t ) is the wealth of agent i at time t and λ ≥ N exchanges. This distribution mechanism is justified byeconomic data in the Appendix of Ref. 20.This model, which we denote as the growth, exchange and distribution (GED) model, wasinvestigated numerically [20] and shown to have a phase transition at λ = 1. For 0 < λ < λ approaches1 from below, the wealth distribution becomes more skewed toward the rich. However, thereis economic mobility and poorer agents can become richer and richer agents can becomepoorer. In addition, every agent’s wealth increases exponentially due to economic growthas the system evolves with time. In contrast, for λ ≥ µ = 0), and there is no economic mobility.Numerical investigations indicate that the phase transition at λ = 1 is continuous [20].The order parameter φ is defined as the fraction of the wealth held by all of the agents exceptthe richest and goes to zero as λ → − (for N → ∞ ). Three exponents were introduced2n Ref. 20 to characterize the behavior of various quantities as λ → − , including the orderparameter φ ∼ (1 − λ ) β and the susceptibility χ ∼ (1 − λ ) − γ (the variance of the orderparameter). As we will discuss in Sec. IV, we can define the total energy of the system andintroduce the exponent α to characterize the critical behavior of the nonanalytic part ofthe mean energy as (1 − λ ) − α . Similarly, we define the specific heat as the variance of theenergy and characterize its divergence as (1 − λ ) − α . Because there is no length scale in theGED model, there is no obvious way of defining a correlation length exponent.Simulations at fixed values of N in Ref. [20] yield the estimates β ≈ γ ≈
1, 1 − α ≈ − α ≈
2. These values do not satisfy the scaling law [21] α + 2 β + γ = 2 , (2)and do not appear to correspond to any known universality class.In this paper we present a mean-field treatment of the GED model and find that theinterpretation of the critical exponents is subtle. The theory shows that if the criticalbehavior is interpreted correctly, the exponents do satisfy Eq. (2) with β = 0, γ = 1 and α = 1. Moreover, we can define an energy and a Hamiltonian that allows us to obtainan equilibrium (Boltzmann) description of the GED model in the limit that the number ofagents N → ∞ . The mean-field theory results are consistent with the simulations [20].In addition to casting light on the nature of the critical point, the mean-field approachpredicts that for λ <
1, the wealth distribution can be made less skewed toward the rich byincreased growth for fixed N , λ , and f . The mean-field approach also indicates that wealthinequality can be reduced for fixed λ < N and µ by decreasing the value of f , corresponding to decreasing the magnitude of the noise. However, for λ ≥
1, economicgrowth does not avoid wealth condensation, and there is no economic mobility.The structure of the remainder of the paper is as follows. In Sec. II we construct an exactdifferential equation for the GED model and then introduce the mean-field approximation tothe equation. In Sec. III we show that there is a phase transition at λ = 1 with critical slowingdown, and obtain the values of the critical exponents β and γ . In Sec. IV we introducethe Ginzburg parameter, define the total energy of the system, and determine the criticalexponent α . In Sec. V we compare the predicted mean-field exponents with the numericalestimates. We discuss the role of multiplicative noise in the GED model in Sec. VI andexamine the relation between the GED model and the geometric random walk in Sec. VII.3inally in Sec. VIII, we discuss the implication of these results for critical phenomena infully connected systems and systems with long but finite-range interactions, and discuss theimplication of our results for the study of economic systems. Because the GED model issimilar in several ways to the fully connected Ising model, we review some aspects of thatmodel in the Appendix and discuss the Ginzburg criterion as a self-consistency check on theapplicability of mean-field theory. II. EXACT AND MEAN-FIELD EQUATIONS
The rate of change of the wealth of agent i is given by a formally exact stochastic differenceequation ∆ w i ( t )1 = f (cid:88) j Θ (cid:2) w i ( t ) − w j ( t ) (cid:3) η ij ( t ) w j ( t )+ f (cid:88) j (cid:110) − Θ (cid:2) w i ( t ) − w j ( t ) (cid:3)(cid:111) η ij ( t ) w i ( t ) + µW ( t ) w i ( t ) λ S ( t ) . (3)The denominator on the left-hand side of Eq. (3) is written as 1 to emphasize that Eq. (3)is a difference equation rather than a differential equation. HereΘ (cid:0) w i − w j (cid:1) = (cid:0) w i ≥ w j (cid:1) (cid:0) w i < w j (cid:1) , (4)and S ( t ) = (cid:88) i w λi ( t ) . (5)The parameter f is the fraction of the poorer agents’s wealth that is exchanged, µ is thefraction of the total wealth that is added after N exchanges, the parameter λ determines thedistribution of the added economic growth, and η ij ( t ) for ı (cid:54) = j is a time-dependent randommatrix element such that η ij ( t ) = i and j do not exchange wealth1 wealth is transferred from agent j to agent i − i to agent j. (6)( η ij = 0 if i = j .) The matrix elements of η can be chosen from any probability distributionwith the constraint that if η ij = ±
1, then η ji = ∓
1. This condition imposes the constraintthat the exchange conserves the total wealth.4o obtain a differential equation we multiply and divide the denominator on the left-handside of Eq. (3) by N , the number of agents. Because we will take the limit N → ∞ and takeone time unit to correspond to N exchanges, we have that 1 /N → dt . Note that in thesimulations of Ref. [20], N exchanges was chosen as the unit of time. In this case each agentwill, on the average, exchange wealth with only one other agent and hence one exchangedescribed by the difference equation would not take place in an infinitesimal amount of time.One exchange per agent does take place in an infinitesimal time if one time unit correspondsto N exchanges during which each agent exchanges wealth with every other agent on theaverage.The parameters f and µ in Eq. (3) are the rates of exchange and growth, respectively,and are defined per N exchanges (to be consistent with the simulations). Hence to obtain aconsistent differential equation, these rates need to be scaled by N . We let f = f /N and µ = µ /N, (7)and assume that f and µ are independent of N . Note that these theoretical considerationsimply that the parameters f and µ in the simulations must be scaled with N if the Ginzburgparameter is held fixed.Because wealth is added to the system after every N exchanges, the total wealth in thesystem at time t is given by W ( t ) = W (0) e µ t . (8)With these considerations Eq. (3) becomes dw i ( t ) dt = f (cid:88) j Θ (cid:2) w i ( t ) − w j ( t ) (cid:3) η ij ( t ) w j ( t )+ f (cid:88) j (cid:110) − Θ (cid:2) w i ( t ) − w j ( t ) (cid:3)(cid:111) η ij ( t ) w i ( t ) + µ W ( t ) w i ( t ) λ S ( t ) . (9)To obtain a mean-field theory, we choose an agent whose wealth is w ( t ) and let w mf ( t )be the mean wealth of the remaining agents. That is, w mf ( t ) = W ( t ) − w ( t ) N − . (10)The mean-field version of Eq. (9) is dw ( t ) dt = f Θ (cid:2) w ( t ) − w mf ( t ) (cid:3) ηw mf ( t ) + f (cid:104) − Θ (cid:2) w ( t ) − w mf ( t ) (cid:3)(cid:105) ηw ( t )+ µ W ( t ) w ( t ) λ S ( t ) . (11)5he quantity S ( t ) defined in Eq. (5) becomes S ( t ) = w λ ( t ) + ( N − w λ mf ( t ) . (12)To obtain a mean-field description we have effectively coarse grained the exchanges be-tween the chosen agent and the remaining N − N exchanges, thechosen agent interacts with N − N exchanges tobe one unit of time in the mean-field theory.It will be convenient to write the growth term in Eq. (11) as µ W ( t ) w λ ( t ) S ( t ) = µ W ( t ) [ w ( t ) /W ( t )] λ [ w ( t ) /W ( t )] λ + ( N − − λ [1 − w ( t ) /W ( t )] λ , (13)where we have used Eqs. (10) and (12) and divided the numerator and denominator by W λ ( t ).To simplify Eq. (11), we first assume that w ( t ) < w mf ( t ); that is, the wealth of the chosenagent is less than the mean wealth of the remaining N − dw ( t ) dt = f η ( t ) w ( t ) + µ W ( t ) [ w ( t ) /W ( t )] λ [ w ( t ) /W ( t )] λ + ( N − − λ [1 − w ( t ) /W ( t )] λ . (14)We divide both sides of Eq. (14) by W and rewrite Eq. (14) as ddt (cid:16) w ( t ) W ( t ) (cid:17) = f η ( t ) w ( t ) W ( t ) + µ [ w ( t ) /W ( t )] λ [ w ( t ) /W ( t )] λ + ( N − − λ [1 − w ( t ) /W ( t )] λ − µ w ( t ) W ( t ) , (15)where we have used the relation [see Eq. (8)]1 W ( t ) dw ( t ) dt = ddt (cid:16) w ( t ) W ( t ) (cid:17) + µ w ( t ) W ( t ) . (16)We next introduce the scaled wealth fraction x ( t ) ≡ w ( t ) W ( t ) , (17)6nd rewrite Eq. (15) as dx ( t ) dt = R ( x, η, t ) (18a)with R ( x, η, t ) ≡ f η ( t ) x ( t ) + µ x ( t ) λ x ( t ) λ + ( N − − λ [1 − x ( t )] λ − µ x ( t ) . (18b)Equation (18) expresses the time-dependence of the wealth of the chosen agent in contactwith a mean-field representing the mean wealth of the remaining agents. Hence, the wealthof the chosen agent is not conserved.For µ = 0, the total wealth W is a constant because the noise η ( t ) that determines thewealth transfer from the mean-field wealth to the chosen agent is the negative of the noisethat governs the wealth transfer from the chosen agent to the mean field.It is easy to show that for zero noise, R ( x, η = 0 , t ) = 0 for x = 0, 1, and 1 /N , and thatthese are the only fixed points of Eq. (18) for λ (cid:54) = 1. To determine the stability of the fixedpoints, we calculate the derivative dR ( x, η = 0 , t ) /dx and obtain dR ( x, , t ) dx = µ λx λ − x λ + ( N − − λ (1 − x ) λ − µ x λ [ λx λ − − λ ( N − − λ (1 − x ) λ − ][ x λ + ( N − − λ (1 − x ) λ ] − µ , (19)where x ≡ x ( t ). For λ <
1, the derivatives at x = 0 and x = 1 are equal to ∞ , which impliesthat these fixed points are unstable. The derivative at x = 1 /N is equal to µ ( λ − x = 1 /N is stable for λ <
1. For λ > x = 1 /N ispositive so that this fixed point is unstable. The derivative at x = 0 and x = 1 equals − w ( t ) > w mf ( t ). The growthterm is the same as before. The exchange term in Eq. (3), f (cid:80) j Θ (cid:0) w i − w j (cid:1) η ij w j , becomes f ηw mf . We use Eq. (10) to write dw ( t ) dt = f η ( t ) W ( t ) − w ( t ) N − µ W ( t ) [ w ( t ) /W ( t )] λ [ w ( t ) /W ( t )] λ + ( N − − λ [1 − w ( t ) /W ( t )] λ . (20)From Eq. (16) and the definition of x ( t ) in Eq. (17) we have dx ( t ) dt = f η ( t ) 1 − x ( t ) N − µ x ( t ) λ x ( t ) λ + ( N − − λ [1 − x ( t )] λ − µ x ( t ) . (21)Equation (18) for the poorer agent and Eq. (21) for the richer agent are the same except forthe noise term, and hence the fixed points are the same. We again use Eq. (10) to rewrite7q. (21) as dx ( t ) dt = f η ( t ) x mf ( t ) + µ x ( t ) λ x ( t ) λ + ( N − − λ [1 − x ( t )] λ − µ x ( t ) . (22)where x mf ( t ) = w mf ( t ) /W ( t ) is the fraction of the mean field agent’s rescaled wealth. Notethat x ( t ) is of order 1 /N as is x mf ( t ). Equation (22) will be used in Sec. III to discuss thephase transition and the critical exponents.In summary, the fixed points for all values of λ are x = 0, 1, and 1 /N for the mean-fieldequations describing the wealth evolution of either the richer or poorer agent. For λ < /N is stable, correspondingto all agents having an equal share of the total wealth on average. For λ >
1, the fixedpoints at 0 and 1 are stable, and the fixed point at x = 1 /N is unstable, which implies thatif all the agents are assigned an equal amount of wealth at t = 0, one agent will eventuallyaccumulate all the wealth in a simulation of the model. Note that if we use the equationfor which the chosen agent is richer than the “mean field” agent, then the stable fixed pointreached when λ > x = 1; similarly, if we chose the equation for which the chosen agentis poorer than the mean field agent, the stable fixed point reached for λ > x = 0. III. THE PHASE TRANSITION
To analyze the phase transition at λ = 1, we investigate Eq. (22), the mean-field differ-ential equation for the richer agent, for x ∼ /N and λ close to 1 − . We let x ( t ) = 1 N − δ ( t ) , (23)assume N δ (cid:28)
1, and expand the second term on the right-hand side of Eq. (18b) to firstorder in
N δ . After some straightforward algebra we find that dδ ( t ) dt = f η ( t ) x mf ( t ) − µ (1 − λ ) δ ( t ) , (24)We multiply both sides of Eq. (24) by N to obtain dN δ ( t ) dt = f η ( t ) ˜ w mf − µ (1 − λ ) N δ ( t ) , (25)where ˜ w mf = N x mf . We write ˜ w mf = 1 + N δ , let φ = N δ, (26)8nd rewrite Eq. (25) as dφ ( t ) dt = f η ( t ) (cid:2) − φ ( t ) (cid:3) − µ (1 − λ ) φ ( t ) . (27)Because φ ( t ) ∼ / √ N (cid:28) N (cid:29)
1, we can ignore φ ( t ) compare to one in Eq. (27) andobtain dφ ( t ) dt = f η ( t ) − (1 − λ ) µ φ ( t ) . (28)The implications of neglecting the term f ηφ ( t ) in Eq. (27), which generates multiplicativenoise, are discussed in Sec. VI. Here we note that the multiplicative noise term vanishes ifthe limit N → ∞ is taken before the critical point is approached, that is, if the mean-fieldlimit is taken before λ →
1. However, for finite N the situation is more subtle.The starting point for the derivation of Eq. (28) was Eq. (22), the mean-field equation forthe richer chosen agent. If the chosen agent is poorer than the average of the other agents,similar arguments lead to the same equation as Eq. (28).The form of Eq. (28) is identical to the linearized version of the Landau-Ginzburg equa-tion [22–24] with φ as the fluctuatng part of the order parameter. Hence, λ = 1 correspondsto a phase transition as was found in simulations of the GED model [20]. As for the usualLandau-Ginzburg equation, the factor of (1 − λ ) sets the time scale for µ (cid:54) = 0. That is, as λ → − , there is critical slowing down, and the system decorrelates on the time scale τ ∼ µ (1 − λ ) . (29)Because the stable fixed point of the poorer agent is zero for λ > λ >
1. Hence,the exponent β , which characterizes the way the order parameter approaches its value atthe transition, is equal to zero.As mentioned, we can assume the noise η ( t ) to be associated with a random Gaussiandistribution of coin flips. Note that η is the average over N coin flips and hence should scaleas √ N /N ∼ / √ N . Hence η ( t ) in Eq. (32) is order 1 / √ N , which implies that φ ( t ) ∼ / √ N and justifies our neglect of terms higher than first order. Simulations in Ref. [20] show thatthe fluctuations are dominated by those near the 1 /N fixed point.To obtain the critical exponent γ , we adopt an approach introduced by Parisi and9ourlas [25] and note that the measure of a random Gaussian noise is given by [25] P ( { η j } ) = exp (cid:104)(cid:82) ∞−∞ − β (cid:80) j η j ( t ) dt (cid:105)(cid:82) (cid:81) j δη j exp (cid:104)(cid:82) ∞−∞ − β (cid:80) j η j ( t ) dt (cid:105) . (30)or P ( η ) = exp (cid:104)(cid:82) ∞−∞ − βN η ( t ) dt (cid:105)(cid:82) δη exp (cid:104)(cid:82) ∞−∞ − βN η ( t ) dt (cid:105) . (31)The factor of N in the argument of the exponential in Eq. (31) comes from the fact that η j ( t ) = η ( t ) for all j in the mean-field approach. This factor of N is consistent with theargument that η ( t ) ∼ / √ N . (In Ref. [26] the factor of N is not explicit, but is implicit inthe integral over all space.)We rewrite Eq. (28) as 1 f dφ ( t ) dt + (1 − λ ) f µ φ ( t ) = η ( t ) , (32)and replace η ( t ) in Eq. (31) by the left-hand side of Eq. (32). This replacement requires aJacobian, but in this mean-field case the Jacobian is unity [23]. Hence the, probability of φ is given by P ( φ ) = exp (cid:110) − βN (cid:82) ∞−∞ (cid:104) f dφ ( t ) dt + µ (1 − λ ) f φ ( t ) (cid:105) dt (cid:111)(cid:82) δφ ( t ) exp (cid:110) − βN (cid:82) ∞−∞ (cid:104) f dφ ( t ) dt + µ (1 − λ ) f φ ( t ) (cid:105) dt (cid:111) . (33)We now assume that the system is in a steady state so that dφ ( t ) /dt = 0 over a timescale of the order of 1 /µ (1 − λ ). Hence, the average (cid:104) φ (cid:105) is given by (cid:104) φ (cid:105) = (cid:82) δφ φ exp (cid:110) − βN (cid:82) ∞−∞ dt (cid:104) µ (1 − λ ) f φ (cid:105) (cid:111)(cid:82) δφ exp (cid:110) − βN (cid:82) ∞−∞ dt (cid:104) µ (1 − λ ) f φ (cid:105) (cid:111) (34)= (cid:82) δφ φ exp (cid:104) − βN µ (1 − λ ) f φ (cid:105)(cid:82) δφ exp (cid:20) − βN µ (1 − λ ) f φ (cid:21) , (35)where the range of integration over time is limited to the interval 1 /µ (1 − λ ).Because we have assumed a steady state, the functional integral becomes a standardintegral over φ . We can take the limits of the integrals to be ±∞ because the factor of10 (cid:29) φ of order 1 / √ N . Hence, Eq. (35) now becomes (cid:104) φ (cid:105) = (cid:82) ∞−∞ dφ φ exp (cid:110) − βN µ (1 − λ ) f φ (cid:111)(cid:82) ∞−∞ dφ exp (cid:110) − βN µ (1 − λ ) f φ (cid:111) . (36)By using simple scaling arguments we see that the second moment of the probability distri-bution diverges as (cid:104) φ (cid:105) ∼ f N µ (1 − λ ) . (37)The fluctuating part of the order parameter φ = N δ is analogous to the fluctuatingpart of the order parameter m = M/N of the fully connected Ising model, where M isthe total magnetization of the system and N is the number of spins. To determine thesusceptibility (per spin) of the Ising model, we need to multiply [ (cid:104) m (cid:105)−(cid:104) m (cid:105) ] by N . Because (cid:104) φ (cid:105) = f [ N µ (1 − λ )] − [see Eq. (37)], the susceptibility (per agent) of the GED model isgiven by χ ∼ f µ (1 − λ ) . (38)We conclude that the susceptibility diverges near the phase transition with the exponent γ = 1.Note that we can relate the variance of φ to the variance of the rescaled wealth. Fromthe definition of δ ( t ) in Eq. (23) and the fact that x ( t ) = w ( t ) /W ( t ) is the rescaled wealth[see Eq. (17)], we have φ ( t ) = 1 − N x ( t ) = 1 − N w ( t ) W ( t ) = 1 − N ˜ w ( t ) . (39)We rescale the total wealth and hence the wealth of each agent so that W ( t ) = N after theincreased wealth due to economic growth has been assigned. Hence ˜ w in Eq. (39) is therescaled wealth of a single agent. Equation (39) will be useful in Sec. V where we comparethe predictions of the theory to the results of the simulations in Ref. 20. IV. THE ENERGY AND SPECIFIC HEAT EXPONENTS
From Eq. (33) we have that P ( φ ) = exp (cid:110) − βN µ (1 − λ ) f φ (cid:111)(cid:82) dφ exp (cid:110) − βN µ (1 − λ ) f φ (cid:111) , (40)11ssuming that the system is in a steady state. From the expression of the action or Hamil-tonian in Eq. (40), where φ is multiplied by βN µ (1 − λ ) /f , we see that the Ginzburgparameter for the GED model is given by (up to numerical factors) G = N µ (1 − λ ) f . (41)To understand why G on Eq. (41) can be interpreted as the Ginzburg parameter, comparethe form of Eq. (40) with the form of the Hamiltonian for the fully connected Ising modelin Eq. (A5) and the dependences of G in Eqs. (41) and (A6) on their respective orderparameters.The inverse temperature β (not to be confused with the order parameter critical ex-ponent), which arises from the amplitude of the Gaussian noise, will be absorbed in theparameter f . The association of β with f is consistent with Eq. (32) in that we arerelating the temperature to the amplitude of the noise and indicates that increasing thefraction of the poorer agent’s wealth transferred in an exchange is equivalent to increasingthe amplitude of the noise.The total energy for the GED model can be seen from the form of the action or theHamiltonian in Eq. (40) E = N φ , (42)in analogy with the Landau-Ginzburg-Wilson free field or Gaussian action for the fullyconnected Ising model [27]. Equations (39) and (42) imply that the total energy of a systemof N agents is given by E = N (cid:88) i =1 (1 − ˜ w i ) (43a)= − N + N (cid:88) i =1 ˜ w i , (43b)where we have used that fact that (cid:80) i ˜ w i = N .The existence of a quantity that can be interpreted as an energy implies that the prob-ability density of the energy is given by the Boltzmann distribution for λ <
1. The latteris consistent with simulations of the GED model [20]. The existence of the Boltzmann dis-tribution also implies that the system is in thermodynamic equilibrium and is not just in asteady state for λ <
1. 12rom Eq. (40) we find that (cid:104) φ (cid:105) ∼ f / [ N µ (1 − λ )]. Hence, we conclude from Eq. (42)that the mean energy per agent of the GED model scales as (cid:104) E (cid:105) N ∼ f N µ (1 − λ ) . (44)Equation (44) suggests that the mean energy per agent diverges as (1 − λ ) − as λ → N , which is not physical. However, if we hold the Ginzburg parameter G constant as λ →
1, we find no divergence (the exponent is zero), which removes the apparent nonphysicalbehavior. That is, (cid:104) E (cid:105) N ∼ (1 − λ ) − (fixed N ) G − (constant G ) . (45)Equation (45) implies that the energy per agent is finite as we approach the critical pointonly if we hold G constant.Near the critical point the nonanalytic behavior of the mean energy per agent can beexpressed as (1 − λ ) − α , where α is the specific heat exponent. Equation (45) for (cid:104) E (cid:105) /N forconstant Ginzburg parameter implies that α = 1. This result for α is what we would find ifwe require that β , γ , and α to satisfy the scaling relation in Eq. (2) with β = 0 and γ = 1.We can also calculate α directly using the probability distribution in Eq. (40). To calculatethe fluctuations in the total energy, we need to calculate the average of φ . If we apply theprobability in Eq. (40), we find that the fluctuations in the energy per agent, and hence thespecific heat is proportional as N f [ µ (1 − λ )] − , where we have multiplied by N as we didfor the susceptibility per spin of the fully connected Ising model. Hence, the specific heat c scales as c ∼ f N µ (1 − λ ) , (46)and c ∼ (1 − λ ) − (fixed N )(1 − λ ) − (constant G ) . (47)We see that if we keep the Ginzburg parameter constant, we find c ∼ f / [ Gµ (1 − λ )] andhence α = 1. Note that if we do not keep G constant, we would find α = 2, which doesnot satisfy Eq. (2). As a consistency check, we can use Eqs. (44) and (46) to construct theGinzburg parameter by comparing the fluctuations of the energy, that is, the heat capacity,13o the mean energy: N c (cid:104) E (cid:105) ∝ f N µ (1 − λ ) = G − . (48) V. COMPARISON WITH SIMULATIONS
The mean-field theory predictions for the exponents α = 1, β = 0, and γ = 1 areconsistent with the simulation results reported in Ref. 20 for fixed G . As discussed inSec. III, mean-field theory also predicts that there is only one time scale near the phasetransition and that the time scale diverges as (1 − λ ) − for fixed Ginzburg parameter, anexample of critical slowing down [see Eq. (29)]. This prediction is consistent with thesimulation results for the mixing time associated with the wealth metric [20] and the energydecorrelation time, which were both found to diverge as (1 − λ ) − for fixed G . The apparentdiscrepancy between the (1 − λ ) − divergence found in the simulations and the (1 − λ ) − divergence predicted by Eq. (29) is due to the difference in the choice of the unit of time in thesimulation ( N exchanges) and the theory ( N exchanges). To account for the difference intime units, we need to divide the simulation result by N with the result that N − (1 − λ ) − ∼ (1 − λ )(1 − λ ) − = (1 − λ ) − , where we have used the relation N ∝ (1 − λ ) − for fixed G [see Eq. (41)].The simulations for fixed G indicate that the energy per agent approaches a constantas (1 − λ ) →
0. This behavior is associated with the nonanalytic part of the energy peragent. This result for the λ -independence of the nonanalytic part of the energy per agentis inconsistent with the relation between the energy per agent and the specific heat, c ∝ ∂ (cid:104) E ( λ ) (cid:105) /∂λ . The (1 − λ ) − dependence of the specific heat for fixed G near λ = 1 suggeststhat the mean energy per agent could include a logarithmic dependence on λ . For example,the form, (cid:104) E (cid:105) /N ∼ a + a L / log(1 − λ ), where a and a L are independent of λ , implies thatthe specific heat scales as c ∼ [log(1 − λ )] − (1 − λ ) − , thus yielding α = 1 with logarithmiccorrections, which standard mean-field theory cannot predict and are very difficult to detectin simulations.There is also agreement between the exponents predicted by mean-field theory and thosedetermined in the simulations when the measurements are done at fixed N . From Eq. (44) wesee that if N is held constant, the mean energy per agent is predicted to diverge as (1 − λ ) − ,which is consistent with the simulations [20], although this divergence, is unphysical because14 Data 33
HP(w)430853037 P ( w ) w λ = 0.7 λ = 0.998 FIG. 1. Comparison of the wealth distribution P ( w ) for λ = 0 . N = 5 × , and M ≈
14 (redcurve) with P ( w ) for λ = 0 . N = 3333, and M ≈
173 (more sharply peaked black points).Both plots are for G = 10 , with f = 0 .
01, and µ = 0 .
1. The two distributions would beidentical if mean-field theory were exact. Plots of P ( w ) for closer values of M [see Eq. (49)] areindistinguishable to the eye. it implies that the mean energy per agent would become infinite. The exponent α is predictedto be two for fixed N , which is also in agreement with the simulations [20]. VI. MULTIPLICATIVE NOISE
A sensitive test of whether the system is in equilibrium is given by the form of the wealthdistribution of the agents. The derivation of the Gaussian form of the wealth distribution[see Eq. (40)] assumes that the system is in a steady state and that G → ∞ and impliesthat the distribution of the energy is a Boltzmann distribution. The wealth distribution P ( w ) in Eq. (40) is predicted to depend only on the value of G and not on the parameters λ , f , and µ separately. Figure 1 shows the distribution of wealth for fixed G = 10 anddifferent values of λ and N . (Values of λ = 0 . λ for fixed G , even though bothdistributions are well fit by a Gaussian. Similar changes in P ( w ) are found for changes inthe other parameters for fixed G .To understand this behavior, we return to Eq. (27), the mean-field equation for theevolution of the wealth near the 1 /N fixed point, which we repeat here for convenience: dφ ( t ) dt = f η ( t ) (cid:2) − φ ( t ) (cid:3) − µ (1 − λ ) φ ( t ) . (27)In Sec. III we argued that the multiplicative noise term f ηφ can be neglected because φ is assumed to be much less than one. However, we retained the “driving” term µ (1 − λ ) φ and did not consider whether the multiplicative noise term was small compared to thedriving term. To determine if this condition holds, we recall that the (average) noise η isassumed to be random Gaussian. Our assumption that the amplitude of the Gaussian noiseis proportional to 1 / √ N is consistent with the dependence of the Gaussian noise in themean-field limit of thermal models such as the Ising model (see, for example, Ref. 26).Because the Gaussian noise is of order 1 / √ N , we can neglect it compared to the drivingterm in Eq. (27) if f / √ N (cid:28) µ (1 − λ ), or M ≡ √ Nf µ (1 − λ ) (cid:29) . (49)Equation (49) defines the parameter M . The condition M (cid:29) G (cid:29) G and M diverge in the mean-field limit for which first N → ∞ and then thecritical point at λ = 1 is approached [28]. If these limits are taken in this order, themean-field treatment neglecting the multiplicative noise is exact (see Ref. [26] andreferences therein).2. A smaller value of f makes the system more describable by a mean-field treatment,which explains the better agreement of the exponents determined from simulationsfor finite values of N , with the exponents calculated from a theory that neglects themultiplicative noise.3. A large value of G does not necessarily imply a large value of M ; that is, as λ →
1, themultiplicative noise can become important even though G is still much greater thanone. 16 f µ τ m τ E τ m and the energy decorrelation time τ E on N , f , and µ for λ = 0 . f and µ are not scaled). Comparison of the first two rows indicatesthat τ m and τ E depend weakly on f for fixed N and µ . Comparison of the second and thirdrows suggests that τ m and τ E depend strongly on the value of µ . Comparison of the third andfourth rows indicates that τ m and τ E are independent of N . These dependencies are in qualitativeagreement with Eq. (29).
4. The Ginzburg parameter G controls the level of mean field and M controls the influenceof the multiplicative noise. It is necessary to keep both parameters constant to obtainresults consistent with the mean-field theory. Because we cannot keep both parametersconstant simultaneously, there will always be some inconsistency of the results for finitevalues of N and G . These inconsistencies can be minimized for sufficiently large N byincreasing µ or decreasing f . The point is that we need to be careful in interpretingthe results of simulations. An example of the limitations of the mean-field theory andthe neglect of both the additive and multiplicative noise terms is shown in Table I.We see that both τ E , the energy decorrelation time, and τ m , the mixing time, dependweakly on f in contrast to Eq. (29) which predicts that these times are independentof f . The dependence of τ E and τ m on f reflects the possible importance of themultiplicative noise. VII. RELATION TO THE GEOMETRIC RANDOM WALK
For either zero growth, µ = 0, or for the critical point, λ = 1, the mean-field equationfor the rescaled wealth, Eq. (18), reduces to dx ( t ) dt = f η ( t ) x ( t ) . (50)17f we use the Ito interpretation for the effect of the multiplicative noise in Eq. (50), thesolution for x ( t ) is x ( t ) = x ( t = 0) exp (cid:104) − f t + f W t (cid:105) , (51)where W t is a Brownian noise or Wiener process and is given by W t = (cid:90) t η ( t (cid:48) ) dt (cid:48) . (52)Because Eq. (50) results from either setting λ = 1 or µ = 0, Eq. (50) implies that the mean-field treatment of the GED model for µ = 0 and λ (cid:54) = 1 results in the same distributionas the geometric random walk without the drift term [29, 30]. For λ = 1 and µ (cid:54) = 0,the solution is e µ t x ( t ), where x ( t ) is the solution with µ = 0, and Eq. (51) describes thedistribution of the geometric random walk with the drift or growth term [29, 30].This result, which follows from the analysis of the mean-field equation, Eq. (18), is notapplicable if N is held constant because G = 0 for µ = 0 or λ = 1, and hence the mean-fieldapproach does not apply. If we keep G constant, Eq. (50) is applicable because the condition G (cid:29) µ ≈ λ ≈ − . To show numerically that the GEDmodel reduces to the geometric random walk at the critical point involves fixing the valueof G and determining the form of the wealth distribution for λ (cid:54) = 1 and µ > λ → µ →
0. Such an extrapolationwould be a difficult and time consuming process.
VIII. SUMMARY AND DISCUSSION
We have investigated a simple agent-based model of the economy in which two agentsare chosen at random to exchange a fraction of the poorer agent’s wealth. Economic growthis distributed according to the parameter λ . The larger the value of λ , the greater thefraction of the growth that is distributed to the agents at the higher end of the wealthdistribution. The model, which we call the GED model, was treated theoretically witha mean-field approach and was shown to have a critical point at λ = 1, consistent withnumerical simulations [20]. The critical exponents are consistent with scaling and the resultsof simulations if the Ginzburg parameter is large and held constant.The agreement of the mean-field theory with the simulations implies that for finite butlarge G and M , the GED model can be characterized as near-mean-field [26]. That is, the18ystem is mean-field in the limit N → ∞ , and is well approximated by mean-field theory ifboth N and M (cid:29)
1, provided that the Ginzburg parameter G (cid:29) • The inclusion of distribution and growth allows the system to be treated by the meth-ods of equilibrium statistical mechanics, but only if the distribution parameter λ < N → ∞ . A similar result holds for modelsof earthquake faults for long-range stress transfer [26, 31]. It is unclear how manydriven dissipative nonequilibrium systems become describable by equilibrium meth-ods in the mean-field limit. If there are systems that do not follow this pattern, whatcharacteristics separate those systems that do from those that do not? • We used equilibrium methods to calculate the critical exponents in agreement with thesimulations, but the exponent α associated with the specific heat is thermodynamicallyconsistent only if the Ginsburg parameter, G , is held constant. Similar results werefound for the fully connected Ising model [27, 32]. Insight into why holding G constantis required for scaling laws to hold in the long-range and fully connected Ising modelis provided by mapping the critical point and spinodal onto a percolation model andinvestigating the geometric structure of the clusters [26]. We do not know of such amapping to a geometric model for the GED model, which raises two related questions.Does such a mapping exists and if so, how could we find it? If the mapping does notexist, what is the physical basis for needing to keep G constant to preserve the scalinglaws? • A subtle feature of using a mean-field approach in treating the GED model for N (cid:29) M defined in Eq. (49). Fromthe agreement of the theory with the simulations, we conclude that the neglect of themultiplicative noise in the theory is a good approximation for M (cid:29)
1. However, unless M → ∞ , the multiplicative noise is always present for finite values of N . Does thepresence of multiplicative noise lead to a breakdown of the equilibrium approximation19y, for example, producing rare events if the system is observed for a sufficientlylong time? The role of multiplicative noise is of particular interest for models of theeconomy in light of the nonergodicity of the geometric random walk, which includesmultiplicative noise [29, 30]. • To have an equilibrium description of critical point behavior, we defined an orderparameter and then obtained the order parameter exponent β and the susceptibilityexponent γ . To obtain the specific heat exponent α , we defined an energy, which alsoallowed us to obtain the λ dependence of the energy as λ approaches its critical value.The definitions of the order parameter and the energy generate a thermodynamicallyconsistent set of exponents that characterizes the critical point. Is our choice of orderparameter and energy unique, or are there other definitions that would lead to anotherset of thermodynamically consistent exponents? • We have considered only a version of the GED model that has no geometry; that is, anagent can exchange wealth with any other agent. An interesting question is whetherour results are altered by including a geometry. We are pursuing this question bysimulating the model on a regular lattice in one and two dimensions as well as onscale free and Erd¨os-Renyi networks. Including a geometry, particularly for regularlattices, will allow us to define a correlation length and investigate the correlationlength exponent ν and the application of hyperscaing ideas and their relation to afixed Ginzburg parameter [26].Any statements about a system as complicated as the economy based on the simpleGED model must be viewed with a considerable amount of caution. However, the resultsobtained from both the numerical and theoretical investigations of the GED model suggestsome general properties of economic systems that are of potential interest. • The form of the exchange term in Eq. (3) assumes that the amount of the exchange isdetermined by the poorer agent. This assumption is a reasonable first approximationbecause in most exchanges of goods or services, the poorer of the two agents decidesif they can afford the exchange. The fact that the wealth transferred is a percentageof the poorer agent’s wealth leads to the multiplicative part of the noise. The amountof the wealth transferred may be interpreted as the difference in value between what20s received by each agent in the exchange. An interesting question is how the natureof the exchange affects the wealth distribution. A more realistic scenario might be tomake the fraction of the poorer agent’s wealth exchanged a stochastic variable. • The exchange term in Eq. (3) also assumes that the winner of the exchange is based onthe toss of a true coin. Such a toss assumes that both agents have equal knowledge ofthe worth of the exchange at the time of the exchange, so that any advantage enjoyedby the winning agent is gained by pure chance. The effect of biasing the coin tossto represent a superior knowledge of either the richer or poorer agent is a subject offuture study. • In the previous paper [20] it was mentioned that the Gini coefficient is not a suitableorder parameter for describing the phase transition in the GED model. The reasonis that for fixed values of λ , f , µ , and N , the wealth distribution, and hence theGini coefficient, is fixed. As a consequence, the Gini coefficient has a variance of zeroin the limit of infinite N or an infinite number of exchanges for finite N , and the“susceptibility,” which would be defined as the variance of the order parameter, wouldbe zero. Although the Gini coefficient fluctuates in numerical simulations of the modelbecause simulations are done for finite N and for a finite number of exchanges, thesefluctuations are not related to the susceptibility. The Gini coefficient does fluctuatein a real economic system because the parameters we have mentioned fluctuate. Thebehavior of the GED model with variable parameters such as λ , f , and µ is a subjectfor further study. • We found that if the distribution of the wealth generated by economic growth is notskewed too heavily toward the wealthy ( λ < λ → − , the wealth distribution becomes more skewed toward thewealthy, thus increasing inequality. The theory indicates that a more unequal distri-bution of added wealth due to growth can be overcome by either increasing the growthparameter µ , decreasing the uncertainty by decreasing f , or by increasing N . Thetheory also indicates that there is a tipping point at λ = 1, so that for λ ≥
1, noincrease of µ or decrease in f can overcome the inequality caused by the distributionof the growth favoring the wealthy. Although the GED model is very simple, this21esult raises the question of whether there is a tipping point in more realistic modelsof the economy. That is, can the distribution of the growth in wealth favor the richto such an extent that the increased wealth (“a rising tide”) is no longer shared bythe majority of people (“lifts all boats”), and the effect of the unequal added wealthdistribution cannot be alleviated by increased growth or decreased uncertainty? • The theory suggests that as the number of agents N is increased, with the parame-ters λ , f , and µ held fixed, the system becomes more describable by a mean-fieldapproach. This result suggests that as globalization increases, mean-field models ofthe global economy might become more relevant and equilibrium methods might bemore appropriate in contrast to economic models that are not ergodic [29, 30, 33]. Westress that an equilibrium treatment would be an approximation and be exact only for N → ∞ , but might be a good approximation for N (cid:29)
1, assuming that G and M areboth much greater than one. The question of how the multiplicative noise would affectthe system if simulated for a very long time is not clear. We found that if the effect ofthe multiplicative noise is increased by lowering M , the wealth distribution developsa tail for large wealth, indicating that the multiplicative noise induces greater wealthinequality. • The model also suggests that increasing the noise amplitude f increases wealth in-equality. In addition, the theory assumes that the parameters λ , f , and growth µ areindependent. These parameters are not necessarily independent in actual economies,which raises the question of how these variables affect each other. For example, µ could be made to depend on λ . If µ is increased as λ is increased, this dependencewould be a test (in the model) of the trickle down theory.Besides the areas of future research raised by these questions, other areas include in-vestigating the effect of growth in models on various network topologies and investigatingdifferent exchange mechanisms and how they affect the distribution of wealth when growthis added. 22 ppendix A: Appendix: The Fully Connected Ising Model It is useful to discuss the analogous equilibrium behavior of the fully connected Isingmodel. To do so, we first consider the long-range Ising model with interaction range R inthe limit that R → ∞ .The Landau-Ginzburg-Wilson Hamiltonian for the long-range Ising model in zero mag-netic field is given by [26] H (cid:0) φ ( (cid:126)y ) (cid:1) = (cid:90) d(cid:126)y (cid:104) R ( ∇ φ ( (cid:126)y )) + (cid:15)φ ( (cid:126)y ) + φ ( (cid:126)y ) (cid:105) . (A1)The integral in Eq. (A1) is over all space, (cid:15) = ( T − T c ) /T c , T c is the critical temperature,and φ ( (cid:126)y ) is the coarse grained magnetization.Near the mean-field critical point we scale φ ( (cid:126)y ) by (cid:15) / and scale all lengths by R(cid:15) − / and obtain H (cid:0) ( ψ ( (cid:126)x ) (cid:1) = R d (cid:15) − d/ (cid:90) d(cid:126)x (cid:104) ( ∇ ψ ( (cid:126)x ) + ψ ( (cid:126)x ) + ψ ( (cid:126)x ) (cid:105) , (A2)where (cid:15) / ψ ( (cid:126)x ) = φ ( (cid:126)y/R ) and (cid:126)x = (cid:126)y/R(cid:15) − / . The integral is over the volume in scaledcoordinates. Because the functional integral over ψ ( (cid:126)x ) is damped for larger values of ψ due to the Boltzmann factor e − βH ( ψ ( (cid:126)x )) and R d (cid:15) − d/ (cid:29)
1, the rescaled magnetization ψ ( (cid:126)x )satisfies the condition, ψ ( (cid:126)x ) < (cid:114) R d (cid:15) − d/ . (A3)For the fully connected Ising model we can ignore the gradient term in H and take R d (cid:15) − d/ → N (cid:15) , and the integral in Eq. (A2) becomes of order one.To calculate the exponent β for the fully connected Ising model, we take (cid:15) < H (cid:0) ψ (cid:1) = N (cid:15) (cid:2) − ψ + ψ (cid:3) . (A4)The most probable value of ψ is obtained by setting the derivative with respect to ψ of H ( ψ )equal to zero, and we find that the most probable value of ψ is ∼ (cid:15) / and β = 1 / χ for the fully connected Ising model, we canignore the quadratic term in Eq. (A4) and write the action of Hamiltonian as H ( ψ ) = N (cid:15) ψ . (A5)We determine the probability as a function of ψ and multiply the average of (cid:15)ψ ( φ ) by N to obtain χ ∼ (cid:15) − as expected. 23ote that the action in Eq. (A5) is order one for the range of fluctuations in the fullyconnected Ising model; that is, ψ < ∼ / √ N (cid:15) . Similarly, we expect the action in the GEDmodel to also be of order one, not order N .The energy per spin of the fully connected Ising model is the square of the magnetizationper spin [26]. Hence the mean energy per spin is the average of (cid:15)ψ = 1 /N (cid:15) . This dependenceon (cid:15) seems nonphysical and seems to imply that the energy per spin diverges as (cid:15) →
0. Tounderstand this result and to calculate the specific heat, we introduce the Ginzburg criterion,which is a self-consistency check on the applicability of mean-field theory [26]. For a mean-field theory to be a good description, the fluctuations of the order parameter must be smallcompared to the mean value of the order parameter. This requirement implies that ξ d χξ d φ = 1 G (cid:28) , (A6)where ξ is the correlation length, χ is the susceptibility, and d is the spatial dimension. TheGinzburg parameter G defined by Eq. (A6) must be much greater than one for mean-fieldtheory to be a good approximation. Much numerical and theoretical work has shown that theGinzburg criterion is a good indicator of the appropriateness of a mean-field description [22,26]. It is in this sense that we will use the Ginzburg criterion in the following.Equation (A6) implies that the Ginzburg parameter for the fully connected Ising modelis given by G = N (cid:15) (up to numerical factors). Mean-field theory for the fully connectedIsing model becomes exact if the limit N → ∞ is taken before (cid:15) → (cid:15) decreasesfor fixed N , G decreases, which implies that the system becomes less describable by mean-field theory. To determine the critical exponents for the fully connected Ising model for alarge but finite value of N in a simulation, we need to keep the system at the same levelof mean field, which implies that we must keep G constant. Hence, as (cid:15) →
0, we need toconsider larger and larger values of N . Keeping G constant has the additional consequenceof restoring two exponent scaling, which is missing in the standard treatments of mean-fieldsystems [27, 34].Another conclusion that follows from the Ginzburg criterion is that the scaling of theisothermal susceptibility χ must be the same as the scaling of ξ d φ or N φ in the fullyconnected Ising model, which justifies multiplying the square of the average of φ = (cid:15) / ψ by N to obtain χ .Because we need to hold G = N (cid:15) constant to find consistent results for the mean-field24sing exponents, the result that the mean energy per spin is proportional to 1 /N (cid:15) can nowbe properly interpreted. We have (cid:104) E (cid:105) N = 1 N (cid:15) = (cid:15)N (cid:15) = (cid:15)G ∼ (cid:15), (A7)where we have assumed that G is a constant. This result is what is expected from a mean-field calculation because the nonanalytic part of the mean energy per spin should scale as (cid:15) − α , with the mean-field value of the specific heat exponent α = 0.We next calculate the specific heat of the fully connected Ising model by recasting theGinzburg criterion in terms of the energy fluctuations. For mean-field theory to be applica-ble, the fluctuations of the energy must be small compared to the square of the mean energy,or ξ d cξ d e = cN (cid:15) = cG → c, (A8)where c is the specific heat. As expected, holding G constant implies that the specific heatexponent α = 0. If we hold N rather than G constant, we would obtain 1 − α = − α = −
2. We see that the two results for α are not consistent unless G is held constant.Note that the exponents β = 1 / γ = 1 are the same whether we hold N or G constant, but the value of α depends on whether N or G is held constant. Also the scalingrelation (2) cannot be satisfied for γ = 1 and β = 1 / α = 0 which in turn impliesthat we need to keep G constant (and large) to obtain a consistent mean-field description.Also note that the form of the right-hand side of Eq. (A5) is the same as the action orHamiltonian that we derived for the GED model using the Parisi-Sourlas method with (cid:15) replaced by µ (1 − λ ) /f [see Eq. (42)]. ACKNOWLEDGMENTS
We thank Ole Peters, Jon Machta, Jan Tobochnik, Bruce Boghosian, and Alex Adamoufor useful discussions. WK would like to acknowledge the hospitality of the London Mathe-matical Laboratory where part of this work was done. [1] J. Angle, “Deriving the size distribution of personal wealth from ‘the rich get richer, the poorget poorer’,” J. Math. Sociol. , 27 (1993).
2] S. Ispolatov, P. L. Krapivsky and S. Redner, “Wealth distributions in asset exchange models,”Eur. Phys. J. B , 267 (1998).[3] A. Dr˜agulescu and V. M. Yakovenko, “Statistical mechanics of money,” Eur. Phys. J B ,723 (2000).[4] J.-P. Bouchaud and M. M´ezard, “Wealth condensation in a simple model of economy,” PhysicaA , 536 (2000).[5] A. Chakraborti and B. K. Chakrabarti, “Statistical mechanics of money: how saving propen-sity affects its distribution,” Eur. Phys. J. B , 167 (2000).[6] A. Chakraborti, “Distributions of money in model markets of economy,” Int J. Mod. Phys. C , 1315 (2002).[7] B. Hayes, “Follow the money,” American Scientist , 400 (2002).[8] Z. Burda, D. Johnston, J. Jurkiewicz, M. Kami´nski, M. A. Nowak, G. Papp, and I. Zahed,“Wealth condensation in Pareto macroeconomies,” Phys. Rev. E , 026102 (2002).[9] J. Angle, “The inequality process as a wealth maximizing process,” Physica , 388 (2006).[10] P. L. Krapivsky and S. Redner, “Wealth distributions in asset exchange models,” Science andCulture , 424 (2010).[11] B. K. Chakrabarti and A. Chakraborti, “Fifteen years of econophysics research,” Sci. Culture , 293 (2010).[12] C. F. Moukarzel, S. Goncalves, J. R. Iglesias, M. Rodriguez-Achach, and R. Huerta-Quintanilla, “Wealth condensation in a multiplicative random asset exchange model,” Eur.Phys. J.-Spec. Top. , 75 (2007).[13] V. M. Yakovenko and J. Barkley Rosser, Jr., “Colloquium: Statistical mechanics of money,wealth, and income,” Rev. Mod. Phys. , 1703 (2009).[14] Bruce M. Boghosian, “The Inescapable Casino,” Scientific American , 70 (2019).[15] B. M. Boghosian, “Kinetics of wealth and the Pareto law,” Phys. Rev. E , 042804 (2014).[16] B. M. Boghosian, A. Devitt-Lee, M. Johnson, J. Li, J. A. Marcq, and H. Wang, “Oligarchyas a phase transition: The effect of wealth-attained advantage in a Fokker-Planck descriptionof asset exchange,” Physica A , 15 (2017).[17] A Devitt-Lee, H. Wang, J. Ii, and B. Boghosian, “A nonstandard description of wealth con-centration in large-scale economies,” Siam J. Appl. Math , 996 (2018).[18] J. Li, B. M. Boghosian, and C. Li, “The Affine Wealth Model: An agent-based model of asset xchange that allows for negative-wealth agents and its empirical validation,” Physica A ,423 (2019).[19] C. Chorro, “A simple probabilistic approach of the Yard-Sale model,” Statistics and Proba-bility Letters , 35 (2016).[20] Kang K. L. Liu, N. Lubbers, W. Klein, J. Tobochnik, B. M. Boghosian, and H. Gould, “Simu-lation of a generalized asset exchange model with economic growth and wealth distribution,”preceding manuscript.[21] J. M. Yeomans, Statistical Mechanics of Phase Transitions , Clarendon Press (1972).[22] S. K. Ma,
Modern Theory of Critical Phenomena , Westview Press (2000).[23] W. Klein and G. Batrouni, “Supersymmetry in spinodal decomposition and continuous order-ing,’ Phys. Rev. Lett. , 1278 (1991).[24] P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod.Phys. , 435 (1977).[25] G. Parisi and N. Sourlas, “Random magnetic fields, supersymmetry, and negative dimensions,”Phys. Rev. Lett. , 744 (1979).[26] W. Klein, H. Gould, N. Gulbahce, J. B. Rundle, and K. Tiampo, “The structure of fluctuationsnear mean-field critical points and spinodals and its implication for physical processes,” Phys.Rev. E , 031114 (2007).[27] L. Colonna-Romano, H. Gould and W. Klein, “Anomalous mean-field behavior of the fullyconnected Ising model,” Phys. Rev. E , 042111 (2014).[28] M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, “On the van der Waals theory of the vapor-liquidequilibrium. I. Discussion of a one-dimensional model,” J. Math. Phys. , 216 (1963).[29] O. Peters, “Optimal leverage from non-ergodicity,” Quant. Fin. , 1593 (2011).[30] O. Peters and W. Klein, “Ergodicity breaking in geometric Brownian motion,” Phys. Rev.Lett. , 100603 (2013).[31] J. B. Rundle, W. Klein, S. Gross and D. L. Turcotte, “Boltzmann fluctuations in numericalsimulations of non-equilibrium threshold systems,” Phys. Rev. Lett. , 1216 (2019).[34] T. S. Ray and W. Klein, “Crossover and breakdown of hyperscaling in long range bond ercolation,” J. Stat. Phys. , 773 (1988)., 773 (1988).