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11 Socio-Economic Modeling
187

In their pioneering paper A. Chakraborty and B.K. Chakrabarthi [3] started
out by stating that the agents taking part in trading exchange their money
according to the rule
∗
∗
v = v + Δ(v, w); w = w − Δ(v, w) . (11.4)
Here Δ(v, w) represents the amount of money to be exchanged, which has to be
such that the agents always keep some money in their hands after trading. The
ratio of saving to all of the money held is usually denoted by s and called the
saving rate. Taking 0 <s< 1 constant, the amount of money to be exchanged
can be modeled as

Δ(v, w) = (1 − s) [(ε −1)v + εw] , (11.5)

where 0 ≤ ε ≤ 1 is a random fraction. This model was further developed in
B.K. Chakrabarthi’s research group by assuming that agents feature a random
saving rate [4]. Clearly, choosing a random value for s does not change the type
of collision events.
A somewhat different trading law was considered by S. Cordier, L. Pareschi
and G. Toscani in [5]. Their trading model reads

∗
∗
v = sv +(1− s)w + ηv ; w = (1 − s)v + sw + ˜ηw , (11.6)
1
where 0 <s< .Here η and ˜η are independent equally distributed random
2
2
variables with variance σ and mean zero. Provided both η and ˜η take values
in the interval [−s, s], the trade (11.6) is such that the random coefficients
p i , q i , i = 1, 2 are nonnegative. Note that this trade is conservative only in the
mean, since p 1 + p 2 = 1+ η = 1, whereas p 1 + p 2 = 1. The last terms in
the trading laws describe the spontaneous growth or decrease of wealth due to
random investments in the stock market and other macro-economic factors.
This mechanism corresponds to the effects of an open market economy where
typically the rich get richer and the poor get poorer.
Non-conservative models have been recently considered by F. Slanina [12],
who introduced a model with increasing total wealth based on the collision
coefficients:

p 1 = s , q 1 = 1− s + ε ; p 2 = 1− s + ε , q 2 = s . (11.7)
In (11.7) ε is a fixed positive constant, so that the total money put into the trade
increases, since

∗
v + w = (1 + ε)(v + w).
∗
This type of trade intends to introduce the feature of a strong economy, which
is such that the total mean wealth is increasing in time. We remark that the