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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 5 (2016) pp 3409-3420
© Research India Publications. http://www.ripublication.com
3409
Statistical Thermodynamics of Money (Thermoney)
Dr. S Prabakaran
Associate Professor School of Economics & Business Administration, Department of Accounting & Finance, Pondificia Universidad Javeriana Cali. Cali, Colombia.
Abstract
Thermodynamics presents an abstract and generalized
approach that enables one to analyze the basic regularities of
various energy processes, even under conditions where the
details of their intrinsic mechanisms are unknown. The
methods of thermodynamics are applicable to systems that
belong to very diverse classes of objects from starts to living
cells. In this work, we attempt to construct the bridge between
statistical thermodynamics and financial side. The main
purpose of this paper to introduce concepts borrowed from
statistical thermodynamics, like entropy into money. The main
goal of this study is fourfold:
1) First we begin our approach through the analogy relation
between statistical thermodynamics and economics
system.
2) Next we demonstrate how the Boltzmann – Gibbs
distribution emerges in economics models.
3) Next we extend this approach through the hypothetical
economic systems in a linear and nonlinear system
4) finally; we construct the thermodynamics monetary system
at Polytropic Constant ( tConsVP nN tan .
and this paper end with conclusion.
Keywords: Statistical Mechanics, Thermodynamic, Money
and Boltzmann – Gibbs distribution, Polytropic Constant.
Historical Introduction The application of statistical physics methods to economics
promises fresh insights into problems traditionally not
associated with physics [1]. Both statistical mechanics and
economics study big ensembles: collections of atoms or
economic agents, respectively.
The fundamental law of equilibrium statistical mechanics is
the Boltzmann-Gibbs law, which states that the probability
distribution of energy is
TCeP
(1)
Where T is the temperature and C is a normalizing
constant [2].
The main ingredient that is essential for the textbook
derivation of the Boltzmann-Gibbs law [2] is the conservation
of energy [3]. Thus, one may generalize that any conserved
quantity in a big statistical system should have an exponential
probability distribution in equilibrium.
An example of such an unconventional Boltzmann-Gibbs law
is the probability distribution of forces experienced by the
beads in a cylinder pressed with an external force [4]. Because
the system is at rest, the total force along the cylinder axis
experienced by each layer of granules is constant and is
randomly distributed among the individual beads. Thus the
conditions are satisfied for the applicability of the Boltzmann-
Gibbs law to the force, rather than energy, and it was indeed
found experimentally [4].
We claim that, in a closed economic system, the total amount
of money is conserved. Thus the equilibrium probability
distribution of money mP should follow the Boltzmann-
Gibbs law Tm
CemP . Here m is money, and T is an
effective temperature equal to the average amount of money
per economic agent. The conservation law of money [5]
reflects their fundamental property that, unlike material
wealth, money (more precisely the fiat, "paper" money) is not
allowed to be manufactured by regular economic agents, but
can only be transferred between agents. Our approach here is
very similar to that of Ispolatovet al. [6]. However, they
considered only models with broken time-reversal symmetry,
for which the Boltzmann-Gibbs law typically does not hold. It
is tempting to identify the money distribution mP with the
distribution of wealth [6]. However, money is only one part of
wealth, the other part being material wealth. Material products
have no conservation law: They can be manufactured,
destroyed, consumed, etc. Moreover, the monetary value of a
material product (the price) is not constant. The same applies
to stocks, which economics textbooks explicitly exclude from
the dentition of money [7]. So, in general, we do not expect
the Boltzmann-Gibbs law for the distribution of wealth. Some
authors believe that wealth is distributed according to a power
law (Pareto-Zipf), which originates from a multiplicative
random process [8]. Such a process may reflect, among other
things, the fluctuations of prices needed to evaluate the
monetary value of material wealth.
Statistical Thermodynamics Examples of economic systems of interest are an individual
consumer or a small country, each of which is embedded
within a larger economic system. Consider an individual
consumer. A fundamental assumption in economics is that the
consumer employs a utility function U to choose to purchase
one good over another. For many purposes, it is sufficient for
the utility to be an ordinal quantity (that is, it specifies only
relative ordering). However, to make the full analogy to
thermodynamics, we must take the utility U to be a real
number. We assume U to be given in a convenient set of
units, such as 1998 dollars, and we also assume that U is
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© Research India Publications. http://www.ripublication.com
3410
measurable [9]. The formalism we develop is falsifiable, and
can be over determined by a proper set of measurements, thus
providing constraints on its consistency. To explain the
analogy, we begin by discussing certain fundamental relations
in economics.
First consider the Measurable Economic Quantity Known as
Wealth,
pNMW (Economics)
(2)
Where and M represents the value and amount of money,
and p and M represents vectors of prices and numbers of
goods.
Economics assumes that the value of an individual
consumer´s money and goods is summarized by the value of
U , which typically exceeds W . The excess is known as the
surplus for which we introduce the notation (psurplus) [10,
11]. Thus
WU (Economics) (3)
In a primitive or very poor economy, there is no surplus, so
0 . In this case, every individual performs the same
economic function at the same efficiency, and there is no
benefit from specialization and trade. The surplus cannot
be negative; for typical economic systems 0 . Although
Equation (3) appears only to define another unknown
quantity, in terms ofU , this economic relationship is
useful because it has a thermodynamic analogue.
The Helmholtz free energy of a system with N identical
particles is defined as [12]
NPVF (Thermodynamics)
(4)
Where P is the pressure, V is the volume, and is the
chemical potential of the particles. We may think of PV
as analogous to M . The quantity in thermodynamics
analogous to the price p is the chemical potential . The
energy E is related to F in terms of the temperature T and
entropy S via
FETS (Thermodynamics)
(5)
Note that, according to the third law of thermodynamics,
0S for a system at 0T
A comparison of Equation (3) and (5) suggests another
analogy, that of handTS . By taking a system with zero
surpluses (and thus zero economic temperature) to have zero
economic entropy, we assume the economic analogue of the
third law of thermodynamics. Here we summarize the
thermodynamic and economic analogies in Table 1.
Table 1: Summary of the suggested analogies between
thermodynamic and economic systems.
THERMO
DYNAMICS F E TS N
ECONOMICS W(Wealth)
U(Utility)
(Surplus)
P(Price)
N (#of
Goods)
Our goal in making an analogy between economics and
thermodynamics is to provide a theoretical framework so that
economics measurements can determine the functional
dependence of the utility U on the economic parameters that
specify the state of an economic system. A knowledge of the
state function as a function of the appropriate economic
parameters completely characterizes the economic system.
From the economic relations we introduce
TS
(6)
And pNMTSWTSU (7)
For E suggests that, from the point of view of its natural set
of variables, we have
NMSUU ,, (8)
Relation (8) is our fundamental assumption.
The economic equivalent is
pdNdMTdSdU (9)
Where
MSNSNM NUp
UU
SUT
,,,
,,
(10)
Let us now apply this theoretical structure
In the Wealth of Nations, Adam Smith distinguishes between
two measures of utility [13]. One measure is the ‘‘value in
exchange.’’ In economics it is conventional to identify the
value in exchange with the price p . From (9), we take this
measure to be the marginal utility per good dNdU at fixed
S and M . Another measure is the ‘‘value in use, ’’ which is
less readily identified. We will identify ‘‘value in use’’ with
the marginal utility per good dNdU for another set of
fixed variables. For simplicity, we will take M to be fixed,
but we cannot be explicit about the second variable that is to
be held fixed, and will simply denote it as x .
From Eq. (9) we then have
pNST
NU
MxMx
,,
(11)
For fixed market values of goods, U be maximized for each
good. Fixed market value means that the goods 1 and 2 are
exchanged in the marketplace subject to the condition [9]
22110 dNpdNp (12)
The maximization of U require that
2
2
1
1
0 dNNUdN
NU
(13)
Combining equation (12) and (13) then yields
tConsNU
ptan
1
(14)
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for each good. Thus, as desired, the ratio of value in use to
price is a constant.
Using equation (11) and (14) can be expressed as
tConsNS
pT
NU
p MxMx
tan11
,,
(15)
From Equation (15) the constancy of this ratio for all goods
does not depend on whether the price is included in the
computation of utility. The present formalism helps us focus
on the issue of ‘‘what is held constant. ’’ Let m represent the
value in use (marginal utility per good, at fixed x and M ):
MxNUm
,
(16)
Note that m is specified in monetary units, from equation (14)
and (15), the ratio pm has the same dimensionless value for
all goods.
The ratio pm can be generalized to include the value of
currency, thus permitting the study of saving. Specifically,
define
NxMUm
,
(17)
Then the ratio of value in use to value in exchange for money,
m , takes on the same value as pm for goods.
If we use equation (7) to relate W and U , the analogy to the
development associated with F leads to
pdNdMSdTdW (18)
Where NWp
MW
TWS
,, (19)
It is implicit that two of the three quantities NMT ,, are
held constant in the partial derivatives. From Equation (18)
we may write the functional dependence
NMTWW ,, (20)
Another standard economic relationship states that when an
individual consumer interacts with the market, the price
(marginal cost per good) is determined by the market. We can
obtain this result by assuming that, in equilibrium, the total
wealth of the consumer and of the market is maximized at
fixed temperature and money. Considering the market to be a
reservoir r , we have from Equation (18).
rrrrrrr dNpdMdTSdW (21)
Subject to the condition 0 rdTdT , conservation of
money 0 rdMdM , and conservation of goods
0 rdNdN , we find by adding equation (18) and (21)
that
dNppdMdWdW rrr (22)
The right-hand side of Equation (22) is zero for arbitrary
variations dM and dN only if the value of money to the
consumer is the same as the value of money to the market:
r .
Similarly, for the value of a good, we have rpp . Note that
equation (2) gives the differential
NdpMdpdNdMdW (23)
The consistency of equation (22) and (18) require that
NdpMdSdT 0 (24)
Equation (24) is the analogue of the Gibbs–Duhemrelation.
Among other things, it implies that a decrease in the price of
money or goods (as when the state of economic development
increases) is accompanied by an increase in the economic
temperature. This qualitative behavior is expected from
conventional economic reasoning. Specifically, if all prices
and currency values are increased by a common factor, then
the system does not really change: By Equation (24) the
temperature is simply increased by the common factor.
However, Equation (24) holds for more general variations.
Note that scaling by a factor of 2 means calling a one-dollar
bill a two-dollar bill, etc. To really change the value of the
dollar would require printing more dollars, which is a real cost
that cannot be scaled away.
Equation (24) has an important application. We can write the
change in the Marshallian surplus, TS , as
NdpMdTdSSdTTdSd (25)
We interpret the term TdS as the change in the economic
value of leisure. To see that Ndp is a surplus, note that the
cost of incrementally purchasing goods (where the first goods
are scarce and, hence, costly) is N
pdN0
where Np , the
price of the Nth good, decreases as N increases (that is,
0dNdp . )However, when purchased all at once, the
actual cost to the consumer is Np , the number of goods times
the latest cost per good. The difference is
Np
p
NNdpNppdN
00 (26)
The difference is positive, because for the limits of integration
in Equation (26), dp is negative. Hence Ndp is the
change in the consumer’s surplus of goods.
Equation (25) shows that there are two types of surplus: the
Veblenian surplus of leisure and the Smithian surplus from
efficiency due to specialization. These ideas are present in
economics, but we are unaware of any previous formal
statement that relates the Marshallian, Veblenian, and
Smithian surpluses. Note that the statement of the constancy
of pm for all goods purchased by a given consumer is the
same as the statement that the ratio of Veblenian surplus per
good to the price per good is a constant for all goods
purchased by a given consumer.
The following ‘‘Maxwell relations’’ are an immediate
consequence of the fact that the order of the cross derivatives
of NMSUU ., does not matter. Thus
MSNM NT
Spor
SNU
NSU
,,
22
(27)
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NSNM MT
Sor
SMU
MSU
,,
22
(28)
NSVS MNpor
MNU
NMU
,,
22
(29)
In economics, these relations are known as Slutsky conditions
[14, 15]. They guarantee that integrals over dU in
MNS ,, space are path independent. Similar Slutsky
conditions can be derived from W , for which the natural
variables are MT , , and N . Equations (27) and (28) are
new; Equation (29) is already known in a form where the
dependence on entropy is not made explicit. The addition of
the variables T and S helps makes more precise the meaning
of the phrase ‘‘all other quantities held constant. ’’
Boltzmann – Gibbs Distribution
Let us consider a system of many economic agents 1N ,
which may be individuals or corporations. In this paper, we
only consider the case where their number is constant. Each
agent i has some money im and may exchange it with other
agents. It is implied that money is used for some economy
activity, such as buying or selling materials products;
however, we are not interested is that aspects. For us the only
result of interaction between agents i and j is that some
money m changes hands [6]:
mmmmmmmm jijiji ,,, ´´ (30)
Notice that the total amount of money is conserved in each
transaction:
ji mm ´´, ji mm (31)
The local conservation law of money [5] is analogous to the
conservation of energy in collisions between atoms.
We assume that the economic system is closed, i. e. there is no
externalflux of money, thus the total amount of money M in
the system is conserved. Also, in the first part of the paper, we
do not permit any debt, so each agent’s money must be non-
negative: 0im . A similar condition applies to the kinetic
energy of atoms: 0i .
Let us introduce the probability distribution function of
money mP , which is defined so that the number of agents
with money between m and dmm is equal to dmmNP
We are interested in the stationary distribution mPcorresponding to the state of thermodynamic equilibrium. In
this state, an individual agent’s moneymi strongly fluctuates,
but the overall probability distribution mP does not change.
The equilibrium distribution function mP can be derived in
the same manner as the equilibrium distribution function of
energy P in physics [2]. Let us divide the system into two
subsystems 1 and 2. Taking into account that money is
conserved and additive:
21 mmm (32)
Whereas the probability is multiplicative:
21PPP (33)
We conclude that
2121 mPmPmmP (34)
The solution of this equation is
Tm
CemP
(35)
Thus the equilibrium probability distribution of money has the
Boltzmann-Gibbs form. From the normalization conditions
0 01 NMdmMmPanddmmP (36)
We find that TC 1 and NMT
Thus, the effective temperature T is the average amount of
money per agent.
The Boltzmann- Gibbs distribution can be also obtained by
maximizing the entropy of money distribution
mInPmPdmS
0 (37)
Under the constraint of money conservation [2].
Following original Boltzmann´s argument, let us divide the
money axis m0 into small bins of size dm and
number the bins consecutively with the index ,...2,1b .
Let us denote the number of agents in a bin b as bN , the total
number being
1b bNN (38)
The agents in the bin b have money mb , and the total money
is
1b bb NmM (39)
The probability of realization of a certain set of occupation
numbers bN is proportional to the number of ways N
agents can be distributed among the bins preserving the set
bN
This number is
!!
!
21 NNN
(40)
The logarithm of probability is entropy
1!!
b bInNInN (41)
When the number bN are big and Stirling´s formula
NInNInN ! applies, the entropy per agents is
1
1
b bbb bb InPP
NInNNNInN
(42)
Where NN
P bb
is the probability that an agent has money bm .
Using the method of Lagrange multipliers to maximize the
entropy S with respect to the occupation numbers bN with
the constraints on the total money M and the total number of
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agents N generates the Boltzmann-Gibbs distribution for
mP [2].
Figure 1: Histogram and points: stationary probability
distribution of money mP . Solid curves: fits to the
Boltzmann- Gibbs law TmmP exp . Vertical lines:
the initial distribution of money.
Theoretical Economic Systems Consider a hypothetical economic system in equilibrium for
which a quantity, say M , is conserved. There is a reasonable
number of arguments [16] which show that certain current
economies can be considered as systems in equilibrium and
some quantities, like the total amount of money in the system,
are conserved during certain periods of time. For the sake of
concreteness we will consider in this work that M is the
conserved money, although our approach can be applied to
any conserved quantity. Suppose that the system is composed
of N agents which compete to acquire a participation m of
M . In real economic systems, the total number of agents is
such a large number that for most applications the limit
N is appropriate. In a closed economic system, the
equilibrium probability distribution (density function) of m is
given by the Boltzmann-Gibbs distribution Tmem ,
whereT is an effective temperature equal to the average
amount of money per agent. The amount of money m that an
agent can earn depends on several additional parameters
l ,..., 21 , which we call microeconomic parameters. Since
the density function can be normalized to 1, we obtain [17]
dexTQxTQ
e TmTm
,,,
(43)
Where xTQ , the partition is function and represents the
set of all microeconomic parameters. Here we have introduced
the notation nxxxx ,..., 21 to denote the possible set of
macroeconomic parameters which can appear after the
integration over the entire domain of definition of the
microeconomic parameters .
Following the standard procedure of statistical
thermodynamics [17], we introduce the concept of mean value
g for any function gg as
dgexTQ
dgg Tm
,
1 (44)
These are the main concepts which are needed in statistical
thermodynamics for the investigation of a system which
depends on the temperature T and macroscopic variables x .
Consider, for instance, the mean value of the money
dmm (45)
And let us compute the total differential md :
dmdmdddmmdmd (46)
The first term of this expression can be further manipulated by
using the definition of the density function (43) in the form
InQInTm (47)
Then, we obtain that 1d and therefore,
0 dd
n
iiidxyTdSmd
1
(48)
Where the entropy S and the ´intensive‘ macroscopic
variables are defined in the standard manner as
dInInS (49)
d
xm
xmy
iii (50)
Clearly, Equation (48) represents the first law of
thermodynamics. Since the definition of temperature and
entropy are in accordance with the concepts of statistical
mechanics, the remaining laws of thermodynamics are also
valid. Similar results can be obtained for any quantity that can
be shown to be conserved in an economic system. This
reflects the fact that different thermodynamic potentials can
be used to describe the same thermodynamic system.
It is useful to calculate explicitly the entropy
dInS (51)
By using the equation (43) in the form
TInQIn (52)
The result can be cast in the form
xTTInQTSmf ,: (53)
So that
ii x
fyTfS
, (54)
This means that the entire information about the
thermodynamic properties of the system is contained in the
expression for the “free money” f which, in turn, is
completely determined by the partition function xTQ , . In
statistical physics this procedure is still used, with excellent
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results, to investigate the properties of thermodynamic
systems. We propose to use a similar approach in
econophysics. In fact, to investigate a model for an economic
system one only needs to formulate the explicit dependence of
any conserved quantity, say money m , in terms of the
microeconomic parameters . From m one calculates
the partition xTQ , function and the free money xTf ,
which contains all the thermodynamic in-formation about the
economic system.
In real economic systems, probably very complicated
expressions for m will appear for which analytical
computations are not available. Nevertheless, the calculation
of the above integrals for the partition function can always be
performed by using numerical methods so that the
corresponding thermodynamic properties of the system can be
found qualitatively.
The determination of the quantity m in terms of the
microeconomic parameters is a task that requires the
knowledge of very specific conditions and relationships
within a given economic system. The first step consists in
identifying the microeconomic parameters which
influences the capacity of an individual agent to compete for a
share m of the conserved quantity M . Then, it is necessary to
establish how this influence should be represented
mathematically so that m becomes a well-defined function of
.
Another aspect that must be considered is the fact that in a
realistic economic system not all agents are equivalent. For
instance, an agent represented by an individual who works at a
factory for a fixed yearly income would be considerably
different from an agent represented by the firm to which the
factory belongs. An important result of econophysics is that
the Botzmann-Gibbs distribution is not affected by the
specific characteristics of the agents involved in the economic
model [18]. For the statistical thermodynamic approach we
are proposing in this work this means that we can decompose
the quantity m into classes ..., III mmm and different
classes can be described by different functions of different
sets of microeconomic parameters. The formalism of
statistical thermodynamics allows us to consider, in principle,
all possible economic configurations as far as m is a well-
defined function of .
In the following subsections we will study several
hypothetical economic systems in which m is given as simple
ordinary functions of the microeconomic parameters. We
expect, however, that these simple examples will find some
applications in the context of economic systems with
sufficiently well-defined microeconomic parameters.
Although the function m can represent any conserved
economic quantity, for the sake of concreteness, we will m
assume that it represents the money and from now will be
referred as to the money function.
The simplest model corresponds to the case
tConscm tan0
Then, the partition function (43) is given by
,...,, 21
0
nT
ceTQ
(55)
Where nid ii ,...,2,1, , represents the
macroeconomic parameters.
The calculation of the thermodynamics variables, according to
Equation (54), yields
iii
TfyInSTIncf
,,0 (56)
Furthermore, from the definition of free money f we
conclude that ,0cm i.e., the mean value of money is a
constant, as expected. This economic model is considerably
simple. Each agent possesses the same amount of money 0c ,
the entropy does not depend on the mean value of the money
0c , and the state equations are ,0cm and Ty ii .
The system is completely homogeneous in the sense that each
agent starts with a given amount of money, 0c , and ends up
with the same amount. Probably, the only way to simulate
such an economic system would be by demanding that agents
do not interchange money; this is not a very realistic situation.
Indeed, the fact that the entropy is a constant, that does not
depend on the mean value of the money, allows us to
renormalize the macroscopic parameters in such a way that
1ii d (57)
for each i , so that the total entropy vanishes.
In this case, from Equation (55) we see that 0cf and the
corresponding equations of state are compatible with the limit
0T . This resembles the argumentation used in the
description of the third law of thermodynamics. This
observation indicates that a completely homogeneous
economic system is not realizable as a consequence of the
third law of thermodynamics.
Consider now the function
11cm (58)
Where 1c is a positive constant. The corresponding partition
function can be written as
0 11
1
1
1111
,cTdedeTQ T
cT
c
(59)
The relevant thermodynamics variables follow equation (53)
and (54)
1111
1,
cTInS
cTTInf (60)
,1,,01
jTyyj
j (61)
And the conservation law (48) becomes
n
j j
jdTTdSmd
2
(62)
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Moreover, comparing the above results with the definition of
f , it can be shown that Tm , and so the fundamentals
thermodynamics equation in the entropic representation [19]
can be written as
n
jjIn
cm
InS21
1 (63)
This expression relates all the extensive variables of the
system and it can be used to derive all the equations of state in
a manner equivalent to that given in Equation (61). Notice that
in this case the entropy is proportional to the temperature
(mean value of money) so that an increase of the average
money per agent is necessarily associated within increase of
entropy. This observation is in agreement with the second law
of thermodynamics. Notice, furthermore, that in a limiting
case, similar to that considered in the first example given
above, it is possible to renormalize the macroeconomic
parameters j such that the last term of the fundamental
equation (63) vanishes. Nevertheless, in order to reach the
minimum value of the entropy it is necessary to consider the
limit 0T . Again, we consider this result as an indication
of the validity of the third law of thermodynamics.
In Equation (59) we have chosen the interval ,0 for the
integration along the variable 1 . As a consequence the
macroscopic variable 1 vanishes from the final expression
for the fundamental equation (62), and consequently 01 y .
However, it is also possible to consider the interval
max
1
min
1 , so that the macroscopic variables min
1 and
max
1 reappear in the fundamental equation and can be used
as extensive variables which enter the conservation law (48).
In a realistic economic system the interval of integration will
depend on the economic significance of the microeconomic
parameter 1 . For the sake of simplicity, we choose in this
work the former case in which the corresponding
macroeconomic parameter does not enter the analysis.
It is interesting to analysis the most general linear system for
which
n
iiiccm
1
0 (64)
Where 1,0 cc , … are positive real constants. It is then straight
forward to calculate thermodynamics variables
,...,1
21
0
nnT
cccccTe
cTQ
(65)
And the relevant thermodynamics variables
,,0 cTInnS
cTTIncf
nn
(66)
.,0 0 nTcmyi (67)
All the macroscopic parameters vanish and the system
depends only on the temperature. However, the total number
of macroscopic parameters n does enter the expression for
the entropy so that to increase the mean value of the money by
the amount
12 TTnm , (68)
it is necessary to increase entropy by an amount
1
2
TTnInS (69)
Both amounts are proportional to the total number of
macroscopic parameters. Another consequence of this analysis
is that once the constants 0c and n are fixed, it is not possible
to change the mean value of the money without changing the
temperature of the system. This means that an isothermal
positive change of m is possible only by increasing the
total amount of money in the system.
Next, consider the quadratic function 2
11cm which
generates the partition function
21
1
21
1
, Tc
TQ
(70)
Where we have considered the microeconomics parameter 1
in the interval , . The corresponding thermodynamics
variables are
,2
,,0,12
11
11
TmTyyIncTInS
jj
(71)
Which can be put together in the fundamental equation
.2
12
1
11
In
cm
InS
(72)
Again we see that the effect of considering the extreme values
of the parameter 1 is that the corresponding macroscopic
variable 1 does not enter the expressions for the
thermodynamic variables and, consequently, the
corresponding intensive thermodynamic variable vanishes.
Furthermore, it is evident that the power of 1 in the money
function leads to a decrease of the mean value m .
To investigate the general case, we consider the monomial
functional dependence
11cm (73)
with being an arbitrary real constant. A straightforward
calculation leads to the following partition function and
thermodynamic variables:
1,
1
1cTTQ (74)
,,1
11
11
TmIncTInS
(75)
and the intensive variables iy are given as in the previous
case. It follows that 0 in order for the mean value of the
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money to be positive. If 1 , the mean value of the money
decreases, whereas it increases for 10 . In such a
hypothetical system, a way to increase the amount of money
per agent would be to identify the microeconomic parameter
1 and apply the measures which are necessary for the money
function m to become 1m with 1 .
If we consider a transition of an economic system from a state
characterized by the parameter 1 to a new state with
parameter 2 , maintaining the same temperature, the mean
value of the money undergoes a change
Tm
12
11
(76)
so that for a positive change we must require that 12 .
Moreover, if we desire a positive change by an amount greater
than the average money per agent Tm , we must
demand that
12 11
Even if the initial state corresponds to a linear system
11 , in which no increase of m is possible, we can
reach a state of greater mean m value by demanding that
212 . Of course, for a positive change of m the "price”
to be paid will result in an increase of entropy by an amount
which is proportional to the coefficient 12 11
Thermodynamics Monetary System Now we might imagine that the value of economic output
flow rate MG in monetary terms will be equated to its price
multiplied by a volume rate of flow zV
ZMM VPG (77)
To reconcile economic concepts with thermo-economic ones
we imagine that at a particular point in time in production the
two output value flowrates are deemed to be equivalent to
each other, though of course they are defined in very different
ways:
0GGM (78)
0G is defined in terms of some measure or complex of
measures of final productive content.
For money system, the ideal economic equation is expressed
as in above equation (78)
MMMZM TkNVP (79)
Where MP is the price level, ZV is output volume, MN is
the number of currency instruments in circulation, Mk is a
nominal monetary standard (£1, €1, $1, etc.) and MT is an
Index of Trading Value, here the velocity of circulation of
money.
This equation is a re-statement of the general quantity theory
of money. The quantity theory is commonly written as
MVPY (80) where P is price level in an economic system, Y is output in
volume terms, M is the quantity of money in circulation and
V is the velocity of circulation. While the left-hand sides of
the equations are comparable, the right-hand side requires
additional clarification. To obtain an exact comparison with
the quantity theory, equation (79) can be written as:
TNkPV (81)
Where Nk is equivalent to money M in circulation and the
index of trading value T in a thermodynamic monetary
system is equivalent to the velocity of circulation V in a
traditional monetary system. The equation has a very similar
format to the ideal gas equation. Pikler[20] has highlighted the
connections between the velocity of circulation and
temperature.
Thus, in equation (79), there are four variables to consider,
and in differential form equation (79), from now on without
the subscripts, can be written as:
TdT
NdN
VdV
PdP
(82)
The nominal currency value k , is fixed by definition, though
it can change its effective value through inflation and
international comparison.
Before developing a thermodynamic representation of a
money system, appoint should be made concerning the
number of variables. In a non-flow thermodynamic system,
both the constant k and the number of molecules N are
fixed, and there are therefore only three variables left to
consider, pressure P , volume V and temperatureT . In a
thermodynamic flow system, on the other hand, there are four
variables left (other than the constant k ):molecular flow N ,
volume flowV , pressure P and temperatureT , but these are
effectively reduced to three, by replacing volume flow V and
molecular flow N with specific volume NVv . Thus
again only three variables remain, enabling thermodynamic
analysis to proceed.
To develop a thermodynamic analysis of a money system, a
similar requirement is necessary; that is, to reduce the four
factors in equation (82) plus k (making five) to three,
preferably without losing output volume flowV .
The probability is that any change in money instrument stock
N may find its way into changes in all three of the other
variables, price, output volume or velocity of circulation,
depending upon the relative elasticity between the three and
with money, and the degree to which a money system is out of
kilter with the stable state, such as the existence of excess
money or high inflation.
Therefore, the preferred arrangement to develop the
thermodynamic characteristics, although not perfect, is to
transpose money units N to the left-hand side of the
equation, and divide output price level P by the number N
of monetary units in circulation to give a Specific Price
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3417
NPPN . Presentations of the inverse PNNP are
also given at each stage however for completeness, as the
results are the same.
The equation (79) could be written as:
kTVPN (83)
Or in the alternative:
kTNV P (84)
By differentiating equation (83) and dividing by kTVPN
we have:
TdT
VdV
PdP
N
N (85)
Where
N
N
PdP
also equals
P
P
NdN
.
Figure 2: Velocity of circulation, specific price, specific
money stock and output volume
Figure 2 illustrates the relationships between the factors. A
change in velocity of circulation T can arise either from a
change in specific price NP or a change in output volumeV ,
or both. And no change in the velocity of circulation T will
occur if a change in specific price NP is balanced by an
equivalent change in output volume V .
Figure 3: Price – Volume Relationships
As illustrated at figure 3, a number of relationships between
price and volume are current in a thermodynamic
presentation.
Figure 4: Specific Price –Output Volume Relationships
Figure 4 illustrates the same relationships, but in terms of
specific price NP .
For an economy where no change in output volume occurs (
V = Constant), increases/decreases in specific price NP are
matched by an equivalent change in the velocity of circulation
T . Such changes in specific price NP involve a change in the
relationship between actual price level P and the money
stock N .
At the other extreme, where no change in the specific price
NP occurs, changes in real output volume V are matched by
appropriate changes in the velocity of circulationT . Should a
movement in money supply N occur, this will be balanced
by a change in price P .
If the velocity of circulation T remains constant, then any
change in output volume V will result in an offsetting change
in the specific price NP , accompanied by a change in the
relationship of price P to money supply N .
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Last, a further relationship between the variables can exist,
which in this book we have called the Polytropic case
tConsVP nN tan ), involving an Elastic Index n , where
changes in all of the factors can take place, but in a complex
manner. This is shown at figure 4.
The polytrophic equation tConsVP nN tan can be
adapted to meet all of the possible processes. A constant
specific price process
0
N
N
PdP
for example is a
polytrophic process with the elastic index n set at zero, and a
constant volume process is one with the elastic index set at
. The iso trading model tConsVPN tan has an
index of 1, with no change in velocity of circulation T .
Because the monetary stock system is effectively one stock
system, though serving all the other sectors, we could use the
polytrophic case to describe its dynamics, since this covers all
the other price-volume relationships.
By combining equation (85) with the polytrophic equation
CVP nN , the following equations describe the polytrophic
case:
VdVn
PdP
VV
PP
N
Nn
N
N
1
2
1
2
(86)
N
Nn
n
N
N
PdP
nn
TdT
PP
TT
11
1
2
1
2
(87)
VdVn
TdT
VV
TT
n
1
1
1
2
1
2
(88)
Thus far, we have not specified how the relationships impact
on the way in which an economy moves and what drives the
relationships, only that changes in one or more of the factors
will be reflected by changes to the others to enable the
equation of state set out at equations (79), (82) and(85) to
balance out.
The above regressions for the two economies of course
average out short-term variations.
Because price P has been divided by money stock N to
compute a specific price NP , and vice-versa for specific
money PN , the relative elasticities between the variables in
the polytrophic case will be different compared to the position
with these variables being separate.
By utilizing the differential form of equation (86) it is possible
to calculate the short-term quarter-on-quarter variation in the
elastic index n for the two economies.
VdVn
NdNand
VdVn
PdP
P
P
N
N
The three charts at figure 5 illustrate the effect of a change in
the elastic index n between specific price NP , output volume
V and velocity of circulationT .
To finish this section, we refer back to the polytrophic
equation (86). Splitting this into its component parts, we have:
tConsVNPVP nn
N tan
(89)
And taking logs and differentiating we have:
VdVn
NdN
PdP
(90)
Figure 5: Elasticity between specific Price NP , Output
Volume V and Velocity of CirculationT .
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3419
Conclusion The paper highlights the analogues of the thermodynamics
axioms to money dynamics. Within this context the idea of a
money system is developed, based on the origin from
thermodynamics. In this work we propose to apply the
standard methods of statistical thermodynamics in order to
investigate the structure and behavior of economic systems.
Here we presented to cover the Boltzmann – Gibbs
distribution of money in thermodynamics of a big ensemble of
economic agents realistic deterministic strategies with money
conservation. From this paper, we introduced concepts
borrowed from statistical thermodynamics, like entropy into
money. The outcomes of this research analyzed started with
and we approached through the analogy relation between
statistical thermodynamics and economics system then we
demonstrated how the Boltzmann – Gibbs distribution
emerges in economics models. Followed by we extended this
approach through the hypothetical economic systems in a
linear and nonlinear system. Finally we constructed the
thermodynamics monetary system at Polytrophic Constant
and this paper ended with conclusion.
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