Conjuncted: Avarice

Greed followed by avarice….We consider the variation in which events occur at a rate equal to the difference in capital of the two traders. That is, an individual is more likely to take capital from a much poorer person rather than from someone of slightly less wealth. For this “avaricious” exchange, the corresponding rate equations are

dck/dt = ck-1j=1k-1(k – 1 – j)cj + ck+1j=k+1(j – k – 1)cj – ckj=1|k – j|cj —– (1)

while the total density obeys,

dN/dt = -c1(1 – N) —– (2)

under the assumption that the total wealth density is set equal to one, ∑kck = 1

These equations can be solved by again applying scaling. For this purpose, it is first expedient to rewrite the rate equation as,

dck/dt = (ck-1 – ck)∑j=1k-1(k – j)cj – ck-1j=1k-1cj + (ck+1 – ck)∑j=k+1(j – k)cj – ck+1j=k+1cj —– (3)

taking the continuum limits

∂c/∂t = ∂c/∂k – N∂/∂k(kc) —— (3)

We now substitute the scaling ansatz,

ck(t) ≅ N2C(x), with x = kN to yield

C(0)[2C + xC′] = (x − 1)C′ + C —– (4)

and

dN/dt = -C(0)N2 —– (5)

Solving the above equations gives N ≅ [C(0)t]−1 and

C(x) = (1 + μ)(1 + μx)−2−1/μ —– (6)

with μ = C(0) − 1. The scaling approach has thus found a family of solutions which are parameterized by μ, and additional information is needed to determine which of these solutions is appropriate for our system. For this purpose, note that equation (6) exhibits different behaviors depending on the sign of μ. When μ > 0, there is an extended non-universal power-law distribution, while for μ = 0 the solution is the pure exponential, C(x) = e−x. These solutions may be rejected because the wealth distribution cannot extend over an unbounded domain if the initial wealth extends over a finite range.

The accessible solutions therefore correspond to −1 < μ < 0, where the distribution is compact and finite, with C(x) ≡ 0 for x ≥ xf = −μ−1. To determine the true solution, let us re-examine the continuum form of the rate equation, equation (3). From naive power counting, the first two terms are asymptotically dominant and they give a propagating front with kf exactly equal to t. Consequently, the scaled location of the front is given by xf = Nkf. Now the result N ≃ [C(0)t]−1 gives xf = 1/C(0). Comparing this expression with the corresponding value from the scaling approach, xf = [1 − C(0)]−1, selects the value C(0) = 1/2. Remarkably, this scaling solution coincides with the Fermi distribution that found for the case of constant interaction rate. Finally, in terms of the unscaled variables k and t, the wealth distribution is

ck(t) = 4/t2, k < t

= 0, k ≥ 0 —– (7)

This discontinuity is smoothed out by diffusive spreading. Another interesting feature is that if the interaction rate is sufficiently greedy, “gelation” occurs, whereby a finite fraction of the total capital is possessed by a single individual. For interaction rates, or kernels K(j, k) between individuals of capital j and k which do not give rise to gelation, the total density typically varies as a power law in time, while for gelling kernels N(t) goes to zero at some finite time. At the border between these regimes N(t) typically decays exponentially in time. We seek a similar transition in behavior for the capital exchange model by considering the rate equation for the density

dN/dt = -c1k=1k(1, k)ck —– (8)

For the family of kernels with K(1, k) ∼ kν as k → ∞, substitution of the scaling ansatz gives N ̇ ∼ −N3−ν. Thus N(t) exhibits a power-law behavior N ∼ t1/(2−ν) for ν < 2 and an exponential behavior for ν = 2. Thus gelation should arise for ν > 2.

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