Consider a population of traders, each of which possesses a certain amount of capital which is assumed to be quantized in units of minimal capital. Taking this latter quantity as the basic unit, the fortune of an individual is restricted to the integers. The wealth of the population evolves by the repeated interaction of random pairs of traders. In each interaction, one unit of capital is transferred between the trading partners. To complete the description, we specify that if a poorest individual (with one unit of capital) loses all remaining capital by virtue of a “loss”, the bankrupt individual is considered to be economically dead and no longer participates in economic activity.
In the following, we consider a specific realization of additive capital exchange, the “random” exchange, where the direction of the capital exchange is independent of the relative capital of the traders. While this rule has little economic basis, the model is completely soluble and thus provides a helpful pedagogical point.
In a random exchange, one unit of capital is exchanged between trading partners as represented by the reaction scheme (j, k) → (j ± 1, k ∓ 1). Let ck(t) be the density of individuals with capital k. within a mean-field description, ck(t) evolves according to
dck(t)/dt = N(t) [ck+1(t) + ck-1(t) – 2ck(t)] —– (1)
with N(t) ≡ M0(t) = ∑k=1∞ ck(t), the population density. The first two terms account for gain in ck(t) due to the interactions (j, k + 1) → (j + 1, k) and (j, k − 1) → (j−1, k), respectively, while the last term accounts for the loss in ck(t) due to the interactions (j, k) → (j±1, k∓1).
By defining a modified time variable,
T = ∫0t dt’N(t’) —– (2)
equation (1) is reduced to the discrete diffusion equation
dck(T)/dT = ck+1(T) + ck-1(T) – 2ck(T) —– (3)
The rate equation for the poorest density has the slightly different form, dc1/dT = c2 − 2c1, but may be written in the same form as equation (3) if we impose the boundary condition c0(T) = 0.
For illustrative purposes, let us assume that initially all individuals have one unit of capital, ck(0) = δk1. The solution to equation (3) subject to these initial and boundary conditions is
ck(T) = e−2T [Ik−1(2T) − Ik+1(2T)] —– (4)
where In denotes the modified Bessel function of order n. consequently, the total density N(t) is
N(T) = e−2T [I0(2T) + I1(2T)] —– (5)
To re-express this exact solution in terms of the physical time t, we first invert equation (2) to obtain t(T) = ∫0T dT′/N(T′), and then eliminate T in favor of t in the solution for ck(T). For simplicity and concreteness, let us consider the long-time limit. From equation (4),
ck(T) ≅ k/√(4πT3) exp (-k2/4T) —– (6)
and from equation (5),
N(T) ≅ (πT)−1/2 —– (7)
Equation (7) also implies t ≅ 2/3 √(πT3) which gives
N(T) ≅ (2/3πt)1/3 —– (8)
ck(t) ≅ k/3t exp [-(π/144)1/3 k2/t2/3] —– (9)
Note that this latter expression may be written in the scaling form ck(t) ∝ N2xe−x2, with the scaling variable x ∝ kN. One can also confirm that the scaling solution represents the basin of attraction for almost all exact solutions. Indeed, for any initial condition with ck(0) decaying faster than k−2, the system reaches the scaling limit ck(t) ∝ N2xe−x2. On the other hand, if ck(0) ∼ k−1−α, with 0 < α < 1, such an initial state converges to an alternative scaling limit which depends on α. These solutions exhibit a slower decay of the total density, N ∼ t−α/(1+α), while the scaling form of the wealth distribution is
ck(t) ∼ N2/αCα(x), x ∝ kN1/α —– (10)
with the scaling function
Cα(x) = e−x2 ∫0∞ du e−u2 sinh(2ux)/u1+α —– (11)
Evaluating the integral by the Laplace method gives an asymptotic distribution which exhibits the same x−1−α as the initial distribution. This anomalous scaling in the solution to the diffusion equation is a direct consequence of the extended initial condition. This latter case is not physically relevant, however, since the extended initial distribution leads to a divergent initial wealth density.