arXiv:1006.5044v1 [q-fin.TR] 25 Jun 2010
Modelling savings behavior of agents in the kinetic exchange models of market
Anindya S. Chakrabarti
Indian Statistical Institute, 203 B. T. Road, Kolkata-700108
(Dated: June 28, 2010)
Kinetic exchange models have been successful in explaining the shape of the income/wealth dis-
tribution in the economies. However, such models usually make some ad-hoc assumptions when it
comes to determining the savings factor. Here, we examine a few models in and out of the domain
of standard neo-classical economics to explain the savings behavior of the agents. A number of
new results are derived and the rest conform with those obtained earlier. Connections are estab-
lished between the reinforcement choice and strategic choice models with the usual kinetic exchange
PACS numbers: 89.65 Gh Economics, Econophysics- 05.20 Kinetic theory- 05.65.+b Self-organized systems
The distributions of income and wealth have long been
found to possess some robust and stable features inde-
pendent of the specific economic, social and political con-
ditions of the economies. Traditionally, the economists
have preferred to model the left tail and the mode of the
distributions of the workers’ incomes with a log-normal
distribution and the heavier right tail with a Pareto dis-
tribution. For a detailed survey of the distributions used
to fit the income and wealth data see Ref. . However,
there have been several studies recently that argue that
the left tail and the mode of the distribution fit well with
the gamma distribution and the right tail of the distribu-
tion follows a power law [2–4]. It has been argued that
this feature might be considered to be a natural law for
economics [2, 5].
The Chakraborti-Chakrabarti  model (CC model
henceforth) can explain the gamma-like distribution
very well whereas the Chatterjee-Chakrabarti-Manna 
model (CCM model henceforth) explains the origin of
the power law. Both models fall in the category of the ki-
netic exchange models. However, the first model assumes
a constant savings propensity and the second model as-
sumes an uniformly distributed savings propensity of the
agents. It may be noted that the distribution of the sav-
ings factor is exogenous in these models. It is not derived
from any optimization on the agent’s part or by some
Ref. considers an exchange economy populated
with agents having a particular type of utility function
and derives the CC model in the settings of a competitive
market. Here, we show that the same methodology could
be applied to derive the CCM model and we can explain
the savings factor accordingly. A further possibility is
also studied where the constancy of the savings propen-
sity over time is relaxed. More specifically, we examine
the cases where the savings propensity is dependent on
the current money holding of the corresponding agent.
 email-id : email@example.com
This model, as we shall see, shows self-organization and
in some cases, it gives rise to bimodality in the money
It is well known the models of utility maximization has
been severely critisized on the grounds of limitations of
computational capability of human beings . Hence,
to model the savings behavior of the agents, we study
some simple thumb-rules and derive the distributions of
savings therefrom and finally the income/wealth distri-
bution. In particular, we consider the savings propensity
as a strategy variable to the agents. Two cases are ex-
plored here. In the first case, the agents take their savings
decisions of their own by reinforcing their choices. In the
second case, the agents adopt the winning strategy. In
all cases, we study the final money distributions.
The plan of this paper is as follows. In section, II we
explain the savings behavior of the agents using argu-
ments from neoclassical economics. In the next section,
we study a self-organizing market where the savings is a
function of the current money holding of the correspond-
ing agent. In section V and VI, we study the savings
behavior of the agents where the savings decision follows
some very simple thumb-rules. Then follows a summary
II.KINETIC EXCHANGE MODELS IN A
By competitive market we mean a market with atomistic
agents who trade with each other knowing that their in-
dividual actions can not possibly influence the market
outcome (that implies there is no strategic interaction be-
tween the agents). We assume that markets are always
cleared by equating supply and demand; that is the mar-
ket is completely free of any friction. Below, we elaborate
on the market structure and the behavior of the agents
A.The Chakraborti-Chakrabarti (CC) Model
This model considers homogenous agents characterized
by a single savings propensity.
derivation of the model following Ref. . The structure
of the economy is the following. It is an exchange econ-
omy populated with N agents each producing a single
perishable commodity. There is complete specialization
in production which means none of the agents produce
the commodity produced by another. Money is not pro-
duced in the economy. All agents are endowed with a
certain amount of money at the very begining of all trad-
ings. Money can be treated as a non-perishable com-
modity which facilitates transactions. All commodities
along with money can enter the utility function of any
agent as arguments. These agents care for their future
consumptions and hence they care about their savings
in the current period as well. It is natural that with
successive tradings their money-holding will change with
time. At each time step, two agents are chosen at ran-
dom to carry out transactions among themselves compet-
itively. We also assume that the preference structure of
the agents are time-dependent that is the parameters of
the utility functions vary over time (Ref. [10, 11]). For
a detailed discussion on the derivation of the resulting
money-transfer equations, see Ref. . Below, we pro-
vide the formal structure and the solution to the model
Let us assume that at time t, agent i and j have been
chosen. Also, assume that agent i produces Qiamount
of commodity i only and agent j produces Qj amount
of commodity j only and the amounts of money in their
possession at time t are mi(t) and mj(t) respectively (for
simplicity, mk(0) = 1 for k=1,2). Notice that the notion
of complete specialization in production process provides
the agents with a reason for trading with each other.
Naturally, at each time step there would be a net transfer
of money from one agent to the other due to trade. Our
focus is on how the amounts money held by the agents
change over time due to the repetition of such a trading
process. For notational convenience, we denote mk(t+1)
as mkand mk(t) as Mk(for k = 1,2).
Utility functions are defined as follows.
Uj(yi,yj,mj) = yαi
jwhere the arguments in both
of the utility functions are consumption of the first (i.e.,
xiand yi) and second good (i.e., xjand yj) and amount
of money in their possession respectively. For simplic-
ity, we assume that the sum of the powers is normal-
ized to 1 i.e., α1+ α2+ λ = 1.
ity prices to be determined in the market be denoted
by pi and pj. Now, we can define the budget con-
straints as follows. For agent i the budget constraint
is pixi+ pjxj+ mi≤ Mi+ piQiand similarly, for agent
j the constraint is piyi+pjyj+mj≤ Mj+pjQj. In this
set-up, we get the market clearing price vector (ˆ pi, ˆ pj) as
ˆ pk= (αk/λ)(Mi+ Mj)/Qkfor k = 1, 2.
By substituting the demand functions of xk, ykand pkfor
We briefly review the
for agent j,
Let the commod-
0 0.5 1 1.5
2 2.5 3
The usual CC model: money distribution among the agents.
Four cases are shown above, viz., λ = 0 (+), λ = 0.2 (×),
λ = 0.5 ( ∗), λ = 0.8 (?). All simulations are done for
O(106) time steps with 100 agents and averaged over O(103)
k = 1, 2 in the money demand functions, we get the most
important equation of money exchange in this model. To
get the final result, we substitute αi/(αi+αj) by ǫ to get
the money evolution equations as
mi(t + 1) = λmi(t) + ǫ(1 − λ)(mi(t) + mj(t))
mj(t + 1) = λmj(t) + (1 − ǫ)(1 − λ)(mi(t) + mj(t))
where mk(t) ≡ Mk and mk(t + 1) ≡ mk (for k= i, j).
Note that for a fixed value of λ, if αiis a random variable
with uniform distribution over the domain [0,1−λ], then
ǫ is also uniformly distributed over the domain [0,1]. For
the limiting value of λ in the utility function (i.e., λ →
0), we get the money transfer equation describing the
random sharing of money without savings.
Interpretation of λ: Here, it is clearly shown that λ
in the CC model is nothing but the power of money in the
utility function of the agents and finally this turns out to
be the fraction of money holding that remains unaffected
by the trading action. However, in this form it can not
be directly interpreted as the propensity to save. Below,
we try to derive λ from an utlity maximization problem
while retaining the kinetic exchange structure and we
show that in this slightly alternative formulation, λ is
indeed the savings propensity as has been postulated.
B.The Chatterjee-Chakrabarti-Manna (CCM)
As is clear from above, in the CC model the savings deci-
sion, the market clearence, the prices are all determined
at the same instant. But the savings decision is usually
made in separation. More specifically, we can model the
savings decision and the market clearence distinctly. The
CCM model takes into account the heterogeneity of the
agents. In particular, it assumes that it is the savings
propensity of the agents which differs from each other.
To derive the same, we assume that the agents take de-
cisions in two steps. First, they decide how much to save
and in the second step, they go to the market with the
rest of the money and take the trading decisions.
Formally, we can analyze a typical agent’s behavior at
any time step t in the following two steps.
(i) Each agent’s problem is to make the decision re-
garding how much to save. For simplicity, we as-
sume that the utility function is of Cobb-Douglas
type. Briefly, at time t the i−th agent’s prob-
lem is to maximize U(ft,ct) = fλi
to ft/(1 + r) + ct = m(t) where f is the amount
of money kept for future consumption, c is the
amount of money to be used for current consump-
tion, m(t) is the amount of money holding at time
t and r is the interest rate prevailing in the market
which can be assumed to be zero in a conservative
framework. This is a standard utility maximization
problem and solving it by Lagrange multiplier, we
get the optimal allocation for the i−th agent as
t= λim(t) and c∗
decision is independent of what other agents are do-
ing. So now the agents will go to the market with
(1 − λi)m(t) only.
t= (1 − λi)m(t). Clearly, this
(ii) Now that each agent has made the savings deci-
sion, they can engage in competitive trade with
each other in the fashion descibed in subsection
IIA with λ → 0 (but λ ?= 0; it is a mathemati-
cal requirement). Note that the amount of money
used by the i-th agent is c∗
The resultant asset exchange equations are those
given by the CCM model .
t= (1 − λi)m(t) only.
mi(t + 1) = λim1(t) + ǫ[(1 − λi)mi(t) +
(1 − λj)mj(t)]
mj(t + 1) = λjm2(t) + (1 − ǫ)[(1 − λi)mi(t) +
(1 − λj)mj(t)]
Interpretation of λi: First we recall the solution of
the savings decision which is f∗
t= λim(t). Note that it
that is λi is nothing but the proportion of money kept
for future usage to the current money holding and this is
by definition the savings propensity.
λ AS A FUNCTION OF MONEY: A
A distinct possibility is that the savings propensity
is a function of money-holding itself. To examine
1 10 100
The usual CCM model: money distribution among the
agents. All simulations are done for O(106) time steps with
100 agents and averaged over O(103) time steps. The
straight line is a guide for a power law with slope -2.
that case, we need λ:[0,∞)→[0,1].
are two possibilities.
an increasing or decreasing function of money-holding.
The simplest forms that we may assume are the following.
The savings propensity can be
(i) λt = c1e−(c2m(t))with c1 <1: Savings propensity
is a decreasing function of money holding. The re-
striction on c1 does not allow any agent to have
savings propensity equals to 1. The system shows
self-organization and assumes a stable probability
density function in the steady state (See Fig. 3
where, for purpose of illustration, c1has been kept
constant at 0.95). It is seen numerically that as
c2increases the distribution converges to an expo-
nential density function. In the other extreme, it
tends to the CC model with λ = c1. For moder-
ate values of c2, the distribution resembles gamma
function. For other values of c1 also, the system
behaves similarly. Note that since c1 is the max-
imum possible savings propensity, for a very low
value of it, the system becomes indistinguishable
from an exponential distribution. While it seems
counter-intuitive that savings propensity falls with
the money holding, this might in fact be possible
since poorer people can not take any chance to gam-
ble whereas richer people can.
(ii) λt= c1(1−e−(c2m(t))) with c1< 1: Savings propen-
sity is an increasing function of money holding. The
economy again organizes itself and the distribution
of money becomes stable over time. However, there
is something more. It is seen numerically that bi-
modality may apper spontaneously in the density
function of money. See Fig. 4. Ref.  discusses
such bimodal distribution of wealth (or money).
There a mixture of the agents was used where two
classes of agents were characterized by two different
and widely separated (but fixed!) savings propen-
0 0.5 1
Savings propensity is negatively related to the level of
money. All simulations are done for O(105) time steps with
100 agents and averaged over O(103) time steps. Here, c1
has been kept constant at 0.95 and c2 has been changed.
The plots include c2= 0.1 (+), 0.5 (×), 1 ( ∗), 2 (?), 4 (?).
It is seen that as c2 increases, the distribution becomes more
and more skewed finally converging to an exponential
0 0.5 1 1.5 2
2.5 3 3.5 4
Savings propensity is positively related to the level of
money. All simulations are done for O(106) time steps with
100 agents and averaged over O(103) time steps. Here,
c1=0.95 and the curves are plotted for c2= 1 (+), 2 (×), 3
( ∗) and 4 (?). Bimodality is clearly seen in the distribution
sities. Such a segregated population gave rise to
bimodal distributions. However, such segregations
are exogenous since the λs are given from outside
the system. Here, however, we have a new model
in which savings decision is completely endogenous
and the economy organizes itself in such a way that
it gives rise to a class of bimodal distributions. It
may be noted that bimodality in the income/wealth
distribution has actually been observed in many
cases (see e.g., ).
It is seen numerically that bimodality appears for
c1 ≥ 0.92 and c2 ≥ 1. Another interesting fea-
ture of this model is that keeping c1 constant as
c2 increases, the monomodal distribution breaks
into a bimodal distribution which again becomes
a monomodal distribution for even larger values of
c2. For example, consider Fig.
maximum value of c2 considered is 4. But as c2
increases further, the distribution again becomes
monomodal. However, it should also be mentioned
that if c1 is too large (for example, if c1 ≥ 0.97),
then the system produces some strange-looking bi-
4 in which the
IV. ‘IRRATIONAL’ DECISION MAKING
The standard economic paradigm of market clearence via
utility maximization has been criticised on the grounds of
limitations of computational capability of human beings
. The main challenge is to derive the homogeneity in
savings behavior of the agents from a very simple thumb-
rule such that the final distribution of income/wealth
looks realistic. A few realistic components of decision-
making are noted in Ref. [14, 15]
(i) Players develop prefernces for choices associated
with better outcomes even though the association
may be coincident, causally spurious, or supersti-
(ii) Decisions are driven by the two simultaneous and
distinct mechanisms of reward and punishment,
which are known to operate ubiquitously in hu-
(iii) Satisficing or persisting in a strategy that yields a
positive but not optimal outcome, is common and
indicates a mechanism of reinforcement rather than
Of particular interest is item (iii) which goes directly
against the derivations stated above (see Section II).
V.FROM ONE TO MANY ...
To model the savings behavior of the agents, we now
make the following assumptions.
(i) The agents do not perform static optimization.
(ii) There is reinforcement in their decision-making
(iii) The agents look for better payoffs. But eventually
each of them converges to a single and simple strat-
egy or thumb rule.
To incorporate the three above-mentioned assumptions,
we model the agents’ saving behavior by Polya’s urn pro-
cess . The model is as follows. Consider the i−th
0 0.1 0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 1
The savings distribution by reinforcement. Four cases are
shown. Case A: a=1, b=1. Case B: a=2, b=2. Case C: a=4,
b=4. Case D: a=4, b=2. The solid and the dotted lines
show the theoretical results (the β(a,b) distributions).
The steady state money distributions for two cases:- (A)
a=1, b=1 and (B) a=4, b=2. Case (A) produces the same
distribution of money as the CCM model (slope -2 in log-log
plot) whereas in case (B) we get a distribution with slope -3
in the log-log plot. The lines drawn have slopes -2 and -3
respectively. All simulations are done for O(106) time steps
with 100 agents and averaged over O(103) time steps.
agent. The choice is binary, he can take any of the two
decisions, to consume (c) or to save (s). His strategies
(c and s) such that c,s = 0,1 and c + s = 1. At each
instant, he chooses the values for c and s. Define Ct(St)
as the number of times c (s) has been assigned a value of
unity in t time periods. The savings propensity at time t
(that is λt) is defined as the ratio of Stto (St+Ct). We
can assume that initially S0= a and C0= b. The rein-
forcement mechanism is incorporated by assuming that
the probability of choosing s = 1 at any time t + 1, is
simply λt. Basically, it is the Polya’s urn model and the
famous result that follows from it is the following (Ref.
). The random variable λtconverge almost surely to
a limit λ. The distribution of λ is β(a,b). (See Fig. 5 ).
0 0.5 1 1.5 2
2.5 3 3.5 4
The steady state money distributions for two cases:- (A)
a = b; a, b→ ∞ and (B) a=4, b=4. Note that case (A) is
identical to CC model with λ = 0.5. All simulations are
done for O(105) time steps with 100 agents and averaged
over O(102) time steps.
(i) (a=b=1): From the above result, λtconverges to λ
where λ ∼ uni[0,1]. The resultant distribution of
money follows a power law. This is the basic CCM
model. See Fig. 6.
(ii) (a=b; a,b→ ∞): λt converges to λ where λ is a
delta function at 0.5. This corresponds to a special
case of the CC model where λ = 0.5. See Fig. 7.
B. For moderate values of a and b
Clearly, (1<a, b<∞): This model gives the gamma-
like part as well as the Pareto tail of the income/money
distribution for different values of a and b. For example,
we show the results of two cases.
(i) (a=4, b=2): The resulting distribution of savings
propensity is clearly β(4,2). The distribution of
money in the steady state follows a power law with
a slope -3 in the log-log plot. See Fig. 6.
(ii) (a=4, b=4): The resulting distribution of savings
propensity is β(4,4). The distribution of money in
the steady state is gamma function-like. See Fig.
VI. FROM MANY TO ONE ...
In Section V, we have discussed how one can derive a
set of distributed savings propensities starting from a
unique value. Here, we discuss the reverse side of the
same coin. We shall show that the agents with different
savings propensities, can converge to a single value over
To model this situation, we assume that the agents treat
savings propensity as a strategy which evolves over time.
A very simple rule of evolution is the following. The
winner in any trade retains his strategy whereas the loser
adopts the winner’s strategy. Note that by winning in a
trading action, we simply imply that the agent who gets
the lion’s share in that particular trading is the winner.
Since it is a relative term, (by refering to Eqn.2) winning
is determined by the value of the stochastic term ǫ. If
ǫ ≥ 0.5, then the i-th agent wins (in Eqn. 2) else the
j-th agent wins. Note that the most important support
of this type of strategy evolution comes from the third
observation by Flache and Macy  noted in Section IV.
Let us assume that the possible saving propensities are
finite and denoted by λ1, λ2, ..., λketc. Also, let us de-
note the number of agents with λisavings propensity at
time t by ni(t) (for i= 1, ..., k). At each time-period two
of the agents are randomly selected and they trade ac-
cording to Eqn. 2 and then the loser adopts the winner’s
savings propensity. This process is repeated untill the
sytem reaches a steady state in terms of savings propen-
sities. After the whole system becomes steady with the
agents with a fixed saving propensity, the system is al-
lowed to evolve further to reach a steady state in terms
A.Convergence in savings propensity
ni(0) = N,(3)
as the total number of agents remains fixed over time
(recall that ni(t) has been defined above as the number
of agents with a particular savings propensity λi). The
agents only shift from one savings propoensity to another
over time. Note that at each (trading) time point, the
number of agents with a particular savings propensity
rises or falls by unity with equal probability (i.e. depend-
ing on whether ǫ ≥ 0.5 or not) or it remains unchanged
if its agents are not selected to trade. To put it formally,
let us assume that the two agents selected two trade have
savings propensities λiand λjrespectively. Then
ni(t + 1) = ni(t) ± 1with equal probability(4)
ni(t + 1) + nj(t + 1) = ni(t) + nj(t) (5)
nk(t + 1) = nk(t) for all k?= i,j. (6)
0 2000 4000 6000 8000 10000 12000 14000 16000
average saving propensity
The convergence in savings propensity. As examples, three
cases are shown. To study the phenomena of convergence in
the savings propensities we focus on the fluctuation of the
average savings propensities. Initially all agents are assigned
with uniformly distributed savings propensities. Hence the
average savings propensity is initially very close to half. But
over time it evolves to reach a steady state where it
maintains a fixed value indicating that all the agents have
the same savings propensity. Thereafter the system behaves
like the usual CC model.
Hence, the number of agents with a particular savings
propensity performs a random walk of unit step and also
note that the walk is bounded below since ni≥ 0 for all
i and also above by Eqn. 3. In fact, the formulation is
akin to the n-players ruin problem  where the random
walk occurs on a simplex given by Eqn. 3. The general n-
players ruin problem is very difficult to solve analytically
for k ≥ 4. Ref.  presents a matrix-theoretic approach
to the problem which reduces the complexity of the com-
putation. However, we are not interested in finding the
exact solutions to the problem. We simply note that the
given enough time the system will ultimately evolve to a
state where there is only one savings propensity and this
is not unique. It can be any of the initial λis (i = 1, 2,,,
k). See Fig. 8.
B. Steady state money distribution
Once all the agents have a single savings propensity,
the system then behaves like the CC model (see Fig. 1).
VII. SUMMARY AND DISCUSSION
The kinetic exchange models have been very successful
in explaining the origin of the gamma function-like dis-
tribution and the power law in the income/wealth distri-
bution. However, these models use the notion of savings
extensively on an ad-hoc basis without offering much the-
oretical understanding of it. The aim of the present paper
is to provide support to the kinetic exchange models by
deriving and explaining them from standard neo-classical
economics paradigm and the not-so-standard models of
reinforcement learning and strategic selection.
Ref.  provides a microeconomic basis for the kinetic
exchange models with homogenous agents. Here, we ex-
tend that model to explain the heterogenous exchange
models where the agents have different savings propensi-
ties (Sec. II). A further possibility is investigated in Sec.
III where the savings propensity of an agent is dependent
on the money holding of that particular agent and hence
it changes over time (as the money holding changes). It
is shown that even in that case, the economy organizes
itself in such a way that the distribution of money be-
comes stable over time. In some cases, the distribution
produces bimodality. Bimodal income/wealth distribu-
tions have indeed been seen in many countries (see e.g.,
However, it is also noted in Sec. IV that the market
clearing, competitive models used extensively in the eco-
nomics literature has been criticised on the grounds of
limitations of computational capability of human beings
(see e.g., ). So we try to explain the kinetic exchange
models assuming that the agents follow some simple rules
of thumb. It is shown that the mechanism of reinforcing
one’s own choice leads to the CCM model  (Sec. V).
The basic result regarding the distribution of the fixed
points follows from the famous Polya’s Urn problem (Ref.
). Next, we show in Sec. VI that the agents following
a simple rule of thumb of selecting the best strategy leads
to the CC model (Ref. ). The game of strategy selec-
tion reduces to the generalized Gambler’s Ruin problem
or the N-player Ruin problem (Ref. ).
The author is grateful to Bikas K. Chakrabarti and
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