# Use of Euler’s theorem in the elucidation of economic concepts in goods
exchange

Juan Villegas-Febres
Grupo de Química Teórica QUIFFIS
Facultad de Ciencias
Universidad de Los Andes
Mérida 5101
Venezuela
<EMAIL_ADDRESS>
Also<EMAIL_ADDRESS>
###### Abstract

Starting from a plausible assumption about the Total Revenue concept, a system
of economic agents, that simulates the exchange of goods, is studied.
Following a methodology equivalent to that used in the statistical-mechanical
determination of the distribution of energies in a physical system, it is
shown that the price of the exchanged goods arises naturally, as well as the
well-known “Law of demand" . It is also shown that by using Euler’s Theorem of
homogeneous functions, the conditions emerge for the appearance of wholesale
and retail prices. We found also that the Total Revenue is the product of two
factors: the configurational entropy and a "free energy" type one. We study
numerically the case in which the exchanged good is money itself, in systems
in which economic agents have a "roof" on the amount of money they can
possess. It is shown numerically that the increase in the quantity of money in
the hands of the agents necessarily leads to its depreciation, and only the
expansion of the economy, understood here as raising the "roof", minimizes
this loss of purchasing power. The concept of indexing is discussed.

_Keywords_ Entropy, Economy, Price, Total Revenue, Indexing

##  1 Introduction

The economy is a transversal issue that affects all areas of human life,
regardless of the objective and subjective conditions of the historical moment
in which we have had to live. As a discipline of study, economics works mostly
with "laws" of an observational type, such as the Law of demand, or the Law of
diminishing returns, fundamentally the product of experience accumulated over
generations. Science, on the contrary, is endowed with very detailed laws,
theorems, real and virtual experiments, the result of the application of the
Scientific Method. One way to give economic studies a scientific basis is to
approach them using, fundamentally, although not necessarily exclusively,
thermodynamics and statistical mechanics, giving rise to what has been called
Econophysics.

A recurring theme has been the distribution of money in a model society, as a
way of understanding the mechanisms that underlie the unequal distribution of
wealth and/or income, observed in real human societies. The simulation
experiments are focused on the study of money exchange, observing, in general,
a distribution of Boltzmann type for the economic agents that earn less income
or wealth and the rest is Pareto type for the higher purchasing power strata.
One aspect that has been explored, in order to seek improvements in socio-
economic inequality, is the effect of limiting the maximum amount of money
and/or wealth that an individual can possess (the so-called “roof" in levels
accessible to the system).

In these studies the money is worked, as is done analogously with energy in
physical systems, under the principle of the “conservation of money" 111That
is, money is neither created nor destroyed in the system, what it does is
transfer from one agent to another economic agent., and the effect of entropy
is considered a side issue. In other words, the description of the economic
system gravitates around a purely “energetic" approach. Studies focused on
entropy, however, are essentially conceptual, and do not culminate in
practical applications. In 1948 E. Shannon “discovered " the entropy studying
the conditions for a message to be sent and received through a copper cable.
His highly renowned Information Theory made it possible to study real systems
where the concept of energy was not applicable, or important, or central.
Because, after all, the money or the goods that are exchanged in the economic
processes of buying and selling, due to their intrinsic characteristics, have
become equivalent to energy, but are not energy. The advantage of working with
entropy is that this is a property that can be defined and calculated without
the need to define energy.

Another issue that has been widely studied in the literature is the origin and
evolution of prices, as an essential element in the formal description of an
economic system where merchandise is bought and sold. The majority economic
thought tells us that when prices increase then the quantity demanded
decreases, and vice versa (“Law of demand" ). This behavior between the
quantity of goods and prices, from a strictly conceptual point of view, as far
as we know, has not been sufficiently addressed by science. Part of the
motivation of this work is translated into the need to have a clear and simple
method to relate these two quantities, within a coherent theoretical
framework. In this way we could have a way, at least qualitatively, of
“redescovering" the properties that should be observed in real economic
systems when the goods to be exchanged is money itself. In other words, when
the market for buying and selling is the currency of a country, which is
purchased with the currency of another country, there is interest in learning
about the mechanisms that appear when money is printed and released into the
torrent of the economy. And this is the second motivation of this work.

Finally, and as occurs in physical thermodynamic systems, knowing the
equivalence between temperature, or its inverse, with some factor in the
statistical description of the probability of occupation of the levels or
states of the agents, is an essential step to make the macroscopic and
statistical descriptions of these systems are also equivalent.

##  2 Economic Model

The way we will proceed is equivalent to when we are study in statistical
mechanics the probability of finding a system, within a set of systems, with
energy $E_{k}$. Here, however, we will not move in space of the energies
$\\{E_{1},E_{2},E_{3},...E_{k},...E_{m}\\}$, but in the "space of goods"
$\\{\mathbf{Q}_{1},\mathbf{Q}_{1},\mathbf{Q}_{3},...,\mathbf{Q}_{k},...\mathbf{Q}_{m}\\}$,
where each of the N economic agents, which simply we will call agents,
possesses a certain quantity of goods $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$,
$\mathbf{Q}_{3},...,\mathbf{Q}_{k},...\mathbf{Q}_{m}$, product of exchange
between them, which we will consider here simulate the buying-selling process
in a real system. The amounts $\mathbf{Q}_{k}$ belong to the natural numbers,
including zero, and have a maximum value $\mathbf{Q}_{m}$ 222This is the so-
called Upper Limit or ”Roof”.. That is, the space like this created is finite.
And also it is fulfilled $\mathbf{Q}_{1}$ < $\mathbf{Q}_{2}$ <
$\mathbf{Q}_{3}<...<\mathbf{Q}_{k}<...<\mathbf{Q}_{m}$.

We will understand by "goods" o "merchandise" those ones that can be measured,
numbered, accumulated, divided, added and exchanged by human being, like
fruits, vegetables, cereals, gold, oil, currencies (physical or electronic),
cryptocurrencies, stocks corporate, etc., and therefore can be associated with
a price, within a scale, as a reflection of its value.

We will assume that the buying and selling of merchandise is completely
random, and that, as is logical to consider, in this process there is also an
exchange of money. However, we are only interested in the distribution of
merchandise, since in this dynamic the price at which they are traded emerges
naturally. Agents do not seek any benefit, nor minimize and/or maximize any
amount, nor do they follow cost-benefit strategies. It is also reasonable to
consider that here those agents who, at any particular moment, do not own
merchandise should not be excluded from the dynamics, since they can continue
to participate in the economic dynamics by obtaining them later by
appropriation, donation, deceit, persuasion, work, etc

Within the so-called commodities, which are basic merchandise, such as
agricultural products and minerals (gold, iron, etc.), it is possible,
formally, to consider a special type of commodity: money Yakovenko (2013),
despite being a product created by human beings, and in that sense be
different from other consumable goods such as food, fuel, etc. That is, the
money itself (we are considering it of the fiat type, even when money backed
by gold, silver, etc., are not excluded), when being exchanged between the
agents, also generates a purchase-sale price, which may be expressed in
another currency that enjoys greater stability and credibility, or on
commodities such as gold, diamonds, etc.

Let us now characterize the economic system thus constructed: the total number
of goods Q in this economic system is simply

$Q=\sum_{k=1}^{m}\,n_{k}\,\mathbf{\overline{Q}}_{k}\,,$ (1)

where

$\mathbf{\overline{Q}}_{k}=(\mathbf{Q}_{k}/n_{k})\,.$ (2)

$n_{k}$ is the number of agents that owns $\mathbf{Q}_{k}$ goods each, and
$1\leq m\leq N$. $n_{k}$ are also known as Occupation numbers.

It must be true, of course, that the total number of agents in the system is

$N=\sum_{k=1}^{m}n_{k}\,.$ (3)

The total quantity of goods $Q$ and the total number of agents $N$ are fixed
quantities. In the system thus constructed, there is no creation or
destruction of agents or goods.

As an example, let us consider that the goods are bananas. In this society of
10 people who exchange, buying or selling, bananas with each other, the total
number of bananas is, for example, 25. At some given moment, 4 people each
have 1 banana; 3 people each have 2 bananas; 2 people each have 4 bananas; and
1 person has 7 bananas. Then it is fulfilled:
$Q=\sum_{k=1}^{m}\,n_{k}\,\mathbf{\overline{Q}}_{k}=4(4/4)+3(6/3)+2(8/2)+1(7/1)=4(1)+3(2)+2(4)+1(7)=25$.

$N=\sum_{k=1}^{m}\,n_{k}=4+3+2+1=10$.

Note that the total number of agent groups or levels is $m=4$.

##  3 Probability calculation

According to combinatorial theory McQuarrie (1976a), the number of ways $w$ to
distribute $N$ agents among levels $n_{1},n_{2},n_{3},...,n_{m}$ is

$w=\frac{N!}{\prod_{k=1}^{m}\,n_{k}!}\,,$ (4)

where $\Pi_{k=1}^{m}n_{k}!=n_{1}\,!n_{2}!\,n_{3}!\,...\,n_{k}!\,...$ is the
productoria of the $n_{k}!$. The symbol $!$ represents the factorial of the
respective term.

Let’s maximize 333Since $w$ is normally a fabulously large number, it is wise
to maximize $\ln w$ instead of $w$. It is possible to show that the maximum of
$\ln w$ coincides with the maximum of $w$. $\ln w$ with respect to the
$n_{k}$, subject to Eqs. 1 and 3. To do this, we will use the Lagrange
Indeterminate Multipliers Method (MIL) Ibid. (a), which allows maximizing a
function subject to conditions, thus obtaining

$\frac{\partial}{\partial
n_{k}}\Bigl{(}\ln\,w-\alpha\sum_{i=1}^{m}n_{i}-\beta\sum_{i=1}^{m}n_{i}\,\overline{Q}_{i}\Bigr{)}=0;k=1,2,3,..,m\,.$
(5)

Since there are two conditions or restrictions imposed on the function
$\ln\,w$, then there are two constants, $\alpha$ and $\beta$.

Using the above equation, it is found that

$-\ln\,n_{k}-1-\alpha-\beta\overline{Q}_{k}=0\,,$ (6)

where we have used the Stirling approximation Ibid. (b) $\ln x!\approx x\ln
x-x$. $\ln$ is the natural logarithm to base $e$.

If the probability of finding $n_{k}$ agents with number of goods
$\overline{Q}_{k}$ is

$p_{k}\,=\,n_{k}/N\,,$ (7)

then from Eq. 6 we can write

$p_{k}=C\,e^{-\beta\overline{Q}_{k}}\,,$ (8)

where

$C^{\,-1}=\sum_{l=1}^{m}\,e^{-\beta\overline{Q}_{l}}\,.$ (9)

For the reader familiar with statistical mechanics, $p_{k}$ is the probability
that, taking a random system from the set of physical systems, it is in the
energy state $k$, with energy $\overline{E}_{k}$. Similarly, $C$ would be the
inverse of partition function; which ensures that the probabilities are
normalized, $\sum_{k}p_{k}\equiv 1$.

If we focus on any of the systems within the set of physical systems, it is
also possible to show that the probability $p_{k}$ of finding a particle with
energy $\epsilon_{k}$ in a set of $N$ particles at temperature $T$ is
$p_{k}=C\,e^{-\beta{\epsilon}_{k}}$, where the inverse of the normalization
constant is $C^{\,-1}=\sum_{l=1}^{m}\ ,e^{-\beta{\epsilon}_{l}}$. This is the
well-known Boltzmann probability distribution Chandler (1987a).

$\beta$ is a fundamental quantity that can only be determined unambiguously by
comparing it to the physical world. In thermodynamic systems, it can be shown
that $\beta\equiv 1/k_{B}T$ Ibid. (c), where $T$ is the absolute temperature
measured in kelvins, and $k_{B}$ is the Boltzmann’s constant. Identifying
$\beta$ as the inverse of temperature allows us to pass, fluidly, from the
statistical description to the classical description, and vice versa. In
statistical mechanics, using first principles, there is no way to correctly
determine the identity of $\beta$, without appealing to classical
thermodynamics. Starting from this relationship, it is transparent to move
coherently between both descriptions.

In this economic system $\beta$ must also be determined, as we will do later.

##  4 Total revenue in the purchase-sale process

Let us suppose for simplicity, but without losing generality, that this system
of economic agents buys and sells a single type of good. Now consider
plausibly that the Total Revenue $P$ Landsburg (2011a) to be received by the
seller, which is the same amount of money that will be paid to the buyer, is
directly proportional to the total quantity of goods sold $Q$, $P\propto Q$.

Going from a proportionality to an equality, it will be necessary to multiply
by a constant, which we will call $b$. We will call this amount unit price of
the goods.

As the quantity of goods $Q$ cannot be a negative number, then the sign of $P$
is the same sign of $b$, which is usually positive 444In April 2020, an
unprecedented event occurred, when the price of WTI oil registered a negative
value. That specific case, which occurred in the context of global combat
against the Covid-19 virus, demonstrating that, in certain circumstances, the
price of a merchandise could become negative.. Therefore

$P=b\,Q\,.$ (10)

For example, if $10$ bananas are bought and sold, at a price of $0.25$
currency units um each, then the buyer will deliver to the seller $P=b\times
Q=0.25$ $um/unit\times 10\,units=2.5\,um$. If the price $b$ is fixed, that is,
it is independent of the quantity of goods in the purchase and sale, then it
is said that we are in a Perfectly Competitive (PC) market Ibid. (d). If, on
the other hand, the price depends on the amount of merchandise traded, then we
are talking about a Monopolistic Competition (MC) market Ibid. (e) .

The table LABEL:Total-Revenue shows 555This table is equivalente to the
Landsburg teaching example Ibid. (f)., by way of example, the calculation of
Total Revenue $P$ and Marginal Revenue $P_{M}$, understood as the change in
$P$ due to the effect of the change in the amount of goods $Q$, $P_{M}=\Delta
P/\Delta Q$, when a product was bought-sold at certain prices. The general
issue is that, in order to be able to sell larger volumes of goods, the seller
gradually reduces the unit price, so that, for example, when the price is
$80$, and therefore $3$ are sold units of the product, then it is obtained
from the sale $P=$ price $\times$ Quantity of Goods $=80\times 3=240$,
calculated in a certain monetary unit $um$. In the latter case,
$P_{M}=(240-180)/(3-2)=+\,60$. But, if the price is lowered to $30$ in order
to sell more merchandise, in this case $8$, then $P=30\times 8=240$, and the
Marginal Revenue $P_{M}=(240-280)/(8-7)=-\,40$. Clearly in the first case,
from the point of view of the seller, it is a convenient situation, since it
obtains higher income ($P_{M}>0$). In the second case, always from the point
of view of the seller, he is losing money, since the total revenue $P$ has
decreased ($P_{M}<0$), despite selling more merchandise. It is obvious that
the maximum profit is achieved at the maximum of $P$, where $P_{M}=0$. Fig. 1
shows the Total Revenue and the Margial Revenue, as a function of the quantity
of goods sold, according to the table LABEL:Total-Revenue. Note that $P_{M}$
changed sign at the maximum of $P$.

Table 1: Examples of Total Revenue and Marginal Revenue. $Quantity$ | $Price$ | $Total\,Revenue$ | $Marginal\,Revenue$
---|---|---|---
1 | 100 | 100 | +100
2 | 90 | 180 | +80
3 | 80 | 240 | +60
4 | 70 | 280 | +40
5 | 60 | 300 | +20
6 | 50 | 300 | 0
7 | 40 | 280 | -20
8 | 30 | 240 | -40
9 | 20 | 180 | -60

Since we are interested in working with quantities that are independent of a
particular monetary unit $um$, the scale on which we will be measuring $P$, in
its various forms, will be on a scale relative to a monetary unit of reference
$um_{0}$, which we will consider fixed and stable over time. For the case
$um_{0}$ we can write $P_{0}=\mathsf{b}_{0}\,Q_{0}$, where $P_{0}$ is the
Total Revenue in $um_{0}$ of $Q_{0}$ goods, $b_{0}$ is the unit price in
$um_{0}$, and $Q_{0}$ is the quantity of goods, which we will take as the
unit. Since the quantity of goods is a number belonging to the natural
numbers, including zero, in any of the scales - $um$ or $um_{0}$ \- it must
have the same meaning: three bananas are three bananas on both scales, ten red
balls are ten red balls on both scales, etc.

Figure 1: Total Revenue ($\bullet$) and Marginal Revenue ($\ast$) vs Quantity
sold, according to the data in Table LABEL:Total-Revenue.

Let us then take the quotient of the expressions for Total Revenue in $um$ and
$um_{0}$, $(P/P_{0})=(b/b_{0})\,(Q/Q_{0})$. From now on, to simplify the
nomenclature, $(P/P_{0})$ will be called $P$, $(b/b_{0})$ will be called $b$,
$(Q/Q_{0})$ will be called $Q$ . It is then clear that $P$, $b$ and $Q$ are
now dimensionless, and would be measured in $um$ relative to their value in
the reference $um_{0}$. Therefore Eq. 10 is still $P=b\,Q$. Viewed this way,
$b$ is a relative price Ibid. (g).

If we derive Eq. 10 with respect to the number of agents $n_{k}$, keeping the
other $n_{j}$ different from $n_{k}$ constant, it is fulfilled

$\overline{P}_{k}=b\,\overline{Q}_{k}\,,$ (11)

where

$\overline{P}_{k}=\Bigl{(}\partial P/\partial n_{k}\Bigr{)}_{n_{j\neq k}}\,,$
(12) $\overline{Q}_{k}=\Bigl{(}\partial Q/\partial n_{k}\Bigr{)}_{n_{j\neq
k}}\,.$ (13)

$\overline{P}_{k}$ is the Total Revenue when associated with the sale of
$\overline{Q}_{k}$ amount of goods.

If we now take Eq. 11, and calculate the statistical average over all possible
values of $k$, we obtain $\overline{P}=b\,\overline{Q}$ for the average values
of $\overline{P}=P/N$ and $\overline{Q}=Q/N$, where

$\overline{P}=\sum_{k=1}^{m}\,p_{k}\,\overline{P}_{k}\,,$ (14)
$\overline{Q}=\sum_{k=1}^{m}\,p_{k}\,\overline{Q}_{k}\,.$ (15)

$p_{k}=n_{k}/N$ is the probability, given by Eq. 8, $\overline{P}$ and
$\overline{Q}$, is the Total Revenue and the Total Quantity of Goods per
agent, respectively.

Strictly speaking, if we want to do a statistical average over all values of
$k$, Eq. 11 would have to be rewritten as
$\overline{P}_{k}=(b\,\overline{Q})_{k}$, which means that the product
$(b\overline{Q})_{k}$ would have a given value for each $k$. If we write
$(b\,\overline{Q})_{k}\approx b\overline{Q}_{k}$, we are considering that $b$
is a constant and that it is the same for any value $\overline{Q}_{k}$, be it
is it big or small. In other words, the unit price $b$ is the same for any
commodity value, whether it is retail ($\overline{Q}_{k}$ small) or wholesale
($\overline{Q}_{k}$ large).

##  5 Euler theorem for homogeneous functions

Let’s compare the last two expressions with the Euler’s Theorem for
homogeneous functions McQuarrie (1976b); Chandler (1987b), better known in
Physical chemistry as Additivity Rule Castellan (1983) in the determination of
the Partial Molar Quantities Lewis and Randall (1923a)

$X=\sum_{k=1}^{m}n_{k}\,\overline{X}_{k}\,;$ (16)

where an extensive property $X$, Euler function of degree $1$, can be
expressed in terms of the corresponding intensive properties
$\overline{X}_{k}$ 666A characteristic of a intensive property is that it is
constant throughout the entire system, and is obtained by dividing two
extensive properties., Euler function of degree $0$

$\overline{X}_{k}=\Bigl{(}\partial X/\partial n_{k}\Bigr{)}_{n_{j\neq k}}\,.$
(17)

It should be remembered that extensive properties Chandler (1987c) are
additive, so that the total value in a given system is the sum of this
property in each of the $m$ parts into which it is divided. For physical
systems, volume, amount of matter, energy, entropy, etc., are examples of
extensive properties. Intensive properties, on the other hand, cannot be
added, as is the case of temperature, chemical potential, density, etc., and
their value is equal and constant in any of the $m$ parts of the system. It
must also be taken into account that the quantities $\overline{X}_{k}$ can be
positive or negative. For example, in physicochemical systems, the molar
volume of a pure substance (that is, the volume occupied by one mole of
substance) is always positive, but the partial molar volume of that same
substance in a mixture with other substances, it can be positive or negative.
The same is true for the partial molar heat capacity Lewis and Randall
(1923b).

If the values of $\overline{X}_{k}$ are independent of each other, as would
happen, for example, with the molar volume when mixing two gases that behave
in an ideal way, then we can ensure that 777In a mixture the partial molar
volume $\overline{V}_{k}=\Bigl{(}\partial V/\partial n_{k}\Bigr{)}_{n_{j\neq
k}}$ of component $k$, which is the Euler property of degree $0$, is equal to
its molar volume $V_{k}/n_{k}$, only when the volume of each component in the
mixture does not depend on the volume of the other components. Here $V_{k}$ is
the volume occupied by $n_{k}$ moles of component $k$.

$\Bigl{(}\partial X/\partial n_{k}\Bigr{)}_{n_{j\neq
k}}\cong\Bigl{(}X_{k}/n_{k}\Bigr{)}\,.$ (18)

Substituting the previous expression in Eq. 16, it is shown that, as expected,
the extensive property $X$ is equal to the sum of the property in each of the
$m$ parts in which the system is composed, $X=\,\sum_{k=1}^{m}\,X_{k}$. What
is clear is that, in any case, what exactly satisfies the Additivity Rule, Eq.
16, are the Euler functions of degree $0$, $\overline{X}_{k}$. In the case of
the economic system studied here, $P$ and $Q$ are Euler functions of degree
$1$, and $\overline{P}_{k}$ and $\overline{Q}_{k}$ are Euler functions of
degree $0$.

From the probability $p_{k}$, Eq. 8, we can write
$\beta\,\overline{Q}_{k}=-\,\ln p_{k}+\ln C$. Taking now the statistical
average of $\beta\,\overline{Q}_{k}$, we have

$\overline{P}=\,\sum_{k=1}^{m}\,p_{k}\,\beta\,\overline{Q}_{k}\,,$ (19)

$\overline{P}=-\,\sum_{k=1}^{m}\,p_{k}\ln p_{k}+\,\ln C\,.$ (20)

Where we have taken $\sum_{k=1}^{m}\,p_{k}=1$.

For Eq. 19 to be consistent with Eqs. 11 and 14, the constant $\beta$ will be
identified with the unit price $b$

$\beta\equiv b\,.$ (21)

The latter is in line with what was done by Foley (1994), when he identified
${\bf\Pi}$ as the vector of entropic prices, when calculating the probability
$p({\bf x}_{k})=N_{k}/N=Ce^{-{\bf\Pi}\,.\,{\bf x}_{k}}$ to find an agent $k$
in an economic system, which is in a state ${\bf x}_{k}$, where $C$ is the
normalization constant. The vector ${\bf x}_{k}$ can be considered as an
n-dimensional generalization of the energies $\epsilon_{k}$ when studying the
probability distribution $p$ in a system of $N$ molecules,
$p(\epsilon_{k})\propto e^{-\beta\,\epsilon_{k}}$, where the Lagrange
multiplier $\beta$ is inversely proportional to the thermodynamic temperature
$T$.

##  6 Configurational Entropy and Information

In physical systems, the average configurational entropy in reduced units
888This is strictly true only when $N\,\gg\,1$, and for systems of $N$
independent and distinguishable particles. Zupanović and Kuić (2018) made an
interesting proof, concluding that the most probable value of Boltzmann
entropy in reduced units (${S}^{\,*}=\ln\,w$), is $N$ times the Gibbs entropy
in reduced units ($\overline{S}^{\,*}=-\sum_{k}\,p_{k}\,\ln p_{k}$), in the
limit $N\,\gg\,1$. $\overline{S}^{\,*}=S^{\,*}/N$, where ${S}^{\,*}=S/k_{B}$.,
for $N$ agents who distribute a total energy $E,$ distributed in $m$ groups or
levels, it can be written as

$\overline{S}^{\,*}=\,-\,\sum_{k=1}^{m}\,p_{k}\,\ln p_{k}\,.$ (22)

This expression is equivalent, for sufficiently large $N$, to the well-known
Boltzmann equation $\overline{S}_{k}^{\,*}=\ln w/N$, where $w$ is calculated
with Eq. 4. In fact, by applying MIL to $\ln w$, what is found is the
probability $p_{k}$ that maximizes the entropy of the system Ben-Naim (2008a).
In physical systems, this is equivalent to minimizing the energy Shavit and
Gutfinger (2008).

Entropy in physical systems is an extensive, state, emergent, non-mechanical,
and statistical property. But the concept of entropy can be applied to systems
of many bodies where the concept of energy does not exist, is not defined, or
does not need to be used. In economic and social systems, in literary studies,
etc., entropy can be applied and calculated, as a measure of the Quantity of
Information that can be extracted from the system.

In 1948 Shannon extended the Second Law of Thermodynamics, that is, entropy,
to non-physical systems, through his Information Theory Shannon and Weaver
(1980), also known as Communication Theory. It derives the function call
Information $I$ 999The logarithm could have any base, mostly $10$, $e$ or $2$.

$I=\,-\,K\,\sum_{k=1}^{m}\,p_{k}\,\log p_{k}\,,$ (23)

where $K$ is a positive constant, which measures the information, the choice
and the uncertainty of the system. It is clear that $\overline{S}^{\,*}=I$,
where, without loss of generality, we have taken $K=1$ Ben-Naim (2008b) and
logarithm base is $e$.

In this context, entropy, or equivalently Information, can be considered as a
measure of the uncertainty in the description of the economic system presented
here.

##  7 Unit price

Comparing the Eqs. 20 and 22, it is clear that we can write

$\overline{P}\,=\,\overline{S}^{\,*}+\ln C(\overline{Q})\,.$ (24)

It is easy to prove, remembering that $p_{k}=n_{k}/N$, which also hold
101010The term $C(\overline{Q})$ implies that the constant $C$ depends on the
value of $\overline{Q}$. When $C$ is written it means that it takes a single
value for all $\overline{Q}_{k}$.

$\overline{P}_{k}\,=\,\overline{S}_{k}^{\,*}+\ln C\,,$ (25)
$\overline{S}^{\,*}=\,\sum_{k=1}^{m}\,p_{k}\,\overline{S}^{\,*}_{k}\,,$ (26)
$\overline{S}^{\,*}_{k}=\,-\,\ln p_{k}\,.$ (27)

In the different interpretations of the Information given in the literature
Ibid. (h), the term $-\,\log p_{k}$ 111111For the analysis it is irrelevant to
use any base for the logarithm. is a measure of “unlikely" , “surprise" or the
“unexpectedness" , then $\overline{S}^{\,*}=-\,\sum_{k}p_{k}\,\ln p_{k}$ a
measure of the average of improbability, surprise, or unexpectedness, given
number of events or measurements. Also, the term $-\ln p_{k}$ is sometimes
called “ self-information" or “information content" Applebaum (2008a).

From the Eqs. 24 and 25 then we have two equivalent expressions for the unit
price $b$

$b\,=\,(\overline{S}^{\,*}+\ln C(\overline{Q}))/\overline{Q}\,,$ (28)
$b\,=\,(\overline{S}_{k}^{\,*}+\ln C)/\overline{Q}_{k}\,.$ (29)

From the above equations, it is true that 121212This is so because
$\overline{Q}=\Bigl{(}\partial\overline{A}^{\,*}/\partial b\Bigr{)}$.

$b\,=\Bigl{(}\partial\overline{S}^{\,*}/\partial\overline{Q}\Bigr{)}\,,$ (30)
$b\,=\Bigl{(}\partial\overline{S}_{k}^{\,*}/\partial\overline{Q}_{k}\Bigr{)}_{{j\neq
k}}\,.$ (31)

##  8 Thermodynamic Euler functions

Let us compare Eq. 24 with the equation of thermodynamic origin, for a system
of $N$ particles at temperature $T$ and volume $V$

$\overline{E}^{\,*}\,=\overline{S}^{\,*}+\overline{A}^{\,*};$ (32)

where $\overline{E}^{\,*}=E^{\,*}/N$ is the relative energy per particle,
$\overline{A}^{\,*}=A^{\,*}/N$ is the relative Helmnholtz free energy per
particle 131313Where $E^{\,*}=\beta E$ and $A^{\,*}=\beta A$.,
$\overline{S}^{\,*}={S}^{\,*}/N=-\sum_{k=1}^{m}\ ,p_{k}\,\ln p_{k}$ is the
reduced entropy per particle (where ${S}^{\,*}=S/k_{B}$),
$p_{k}=Z^{-1}e^{-\beta\overline{E}_{k}}$ is the probability of finding a
system with energy $\overline{E}_{k}$,
$Z=\sum_{k=1}^{m}\,e^{-\beta\overline{E}_{k}}$ is the partition function, and
$\beta\,=1/k_{B}\,T$. From statistical mechanics it is also known that
Bromberg (1984) $\overline{A}^{\,*}=-\ln Z$.

Since $E^{\,*}$, $S^{\,*}$ and $A^{\,*}$ are Euler functions of degree 1 and
therefore satisfy expressions of the type
$X=\sum_{k=1}^{m}\,n_{k}\,\overline{X}_{k}$, from the previous expression we
can ensure that

$\overline{E}^{\,*}_{k}\,=\overline{S}^{\,*}_{k}+\overline{A}^{\,*}_{k}\,.$
(33)

Therefore, the Average Total Revenue per economic agent $\overline{P}$ is
equivalent, in physical systems, to the energy $\overline{E}^{\,*}$, and has
two contributions: a configurational entropy $\overline{S}^{\,*}$ and another
of the free energy type $\overline{A}^{\,*}=\ln C(\overline{Q})$.

Note that by Euler’s Theorem 16, also $\overline{P}_{k}$ is equivalent to
$\overline{E}^{\,*}_{k}$, and $\ln C$ is equivalent to
$\overline{A}^{\,*}_{k}$.

##  9 Absolute negative temperatures

When studying thermodynamic systems, it is assumed that $\beta=1/k_{B}T$ is
always positive. However, this is not strictly correct for systems where the
energy states accessible to the system have a "roof" of energy $E_{m}$.

Let’s conceive a system where energy levels are limited. The vast majority of
classical systems are not bounded or limited in terms of the maximum energies
that they can acquire. However, there are quantum systems which show this
bounded energy. In this case, and as illustrated schematically in Fig. 2, as
the energy in the system increases, the states are populated of greater energy
until reaching a situation that we can define as “infinite temperature", in
which it is true that the occupancy numbers of the level $k$ tend to the unit
$n_{k}\approx 1$. However, if we continue to increase the energy, and as the
accessible levels are upper-bounded, the particles gradually "accumulate" in
the level of highest energy $E_{m}$.

At $N$ and volume $V$ constant, there is an exact relationship between entropy
$\overline{S}^{\,*}$, energy per particle $\overline{E}=E/N$, and temperature
$T$ Chandler (1987d)

$(k_{B}\,T)^{-1}=\Bigl{(}\partial\overline{S}^{\,*}/\partial\overline{E}\Bigr{)}\,.$
(34)

Figure 2: Representation of a system where the number of energy levels is
limited by a “roof" $E_{m}$.

For the classical case -without “roof"- starting from zero entropy, it is
clear that the entropy will grow monotonically with the energy, so that then
the temperature, and therefore $\beta$, will always be positive. For the
quantum case -with “roof"- the temperature will always be positive, before
reaching “infinite temperature", since the entropy is increases as the energy
increases, as occurs in the classical case, at maximum $T\to\pm\infty$ and
$\beta\to\pm 0$. But then, as the particles “accumulate" at the highest energy
level, the entropy clearly decreases as the energy continues to increase,
until to reach zero entropy again, then the temperature $T$, and therefore
$\beta$, would now be negative.

Figure 3: Schematic representation of the entropy $\overline{S}^{\,*}$ vs the
energy $\overline{E}$ of a thermodynamic system, where the number of energy
levels is finite.

The existence of negative absolute temperatures has a solid theoretical basis
Landau and Lifshitz (1964), being observed experimentally in quantum systems
with a limited number of energy states Purcell and Pound (1951) and
conceptually clarified by Ramsey (1956) in the 50s of the last century. These
types of temperatures have also been observed in experiments as dissimilar as
the ultracold bosons S. Braun et al. (2013) and in the Axelrod model of
cultural convergence Villegas-Febres and Olivares-Rivas (2008).

Figure (3) schematically represents the entropy $\overline{S}^{\,*}$ as a
function of energy $\overline{E}$, for a system with a finite number of
states. The slope, point to point, is $\beta=1/k_{B}\,T$.

One of the consequences of this type of temperature is that the Absolute
temperature scale must also contain negative values:

$+0\,K...+300\,K...+\infty\,K...-\infty\,K...-300\,K...-0\,K\,.$ (35)

Note that the lowest possible temperature is still $0\,K$, but it is preceded
by the symbol “+", that is, the “positive zero", $+0$, to differentiate it
from $-0$, or “negative zero". This $+0$, would be the zero that we are
“accustomed" to working with. Starting from $0\,K$, as we increase the energy
of this system, its temperature reaches higher and higher values (always
positive), until we reach the “plus infinity", $+\infty\,K$. This $+\infty\,K$
would be the infinity we are "used" to working with. From a statistical point
of view, each of the particles in the system now occupies a different energy
state. If we now continue to increase the energy of the system, the particles
will necessarily "accumulate" ("condense") at the maximum energy level (the
“roof" energy). From that moment the temperatures will all begin to be
negative, and therefore the inversion of the population will begin to be
observed and it will begin to drop from $-\,\infty\,K$ 141414The temperature
$-\,\infty\,K$ is identical to $+\,\infty\,K$ and both give the same
distribution and the same values of thermodynamic quantities of the system.
until reaching a hypothetical situation where all the particles would be at
that maximum energy level. There the temperature will return to be zero
kelvin, but negative, $-0\,K$.

That is, negative temperatures are much “warmer" than positive “normal"
temperatures. In other words, the “true zero" , that is, the lowest possible
temperature, is still $+\,0\,K$. The negative zero, $-\,0\,K$, then, turns out
to be much “hotter" than infinity, positive or negative.

Finally, as demonstrated by Dunn and Fosdick (1980): a) Equilibrium states
with negative temperatures are possible. b) Those states with infinite
temperature have the greatest total entropy. c) Those states with negative
temperatures have a greater total energy than any state with positive
temperature.

##  10 Price determination in $k$

Let’s look at $\overline{P}_{k}$, the total revenue per agent, when
$\overline{Q}_{k}$ quantities of merchandise are traded. For example: in a
given $k$, where there are two agents, $n_{k}=2$, a total of 10 USD was paid
for 20 pens. Here, the Total Revenue on the site $k$, $P_{k}=10$. The total
quantity of merchandise $Q_{k}=20$. These two amounts per economic agent are:
$\overline{P}_{k}=10/2=$5 and $\overline{Q}_{k}=20/2=$10\. If we divide 10 by
20 pens, 10 / 20 = 0.5, we get the price per merchandise; that is, the unit
price $b$. So if we divide $\overline{P}_{k}$ by $\overline{Q}_{k}$, we get,
as expected, the unit price $b=\overline{P}_{k}/\overline{Q}_{k}$ .

If we now ask ourselves, statistically, about the probability of finding, in
the entire economic system, a given Total Revenue per agent $\overline{P}_{k}$
(and/or merchandise $\overline{Q}_{k}$), we get the Total Expected Revenue
$\overline{P\,^{\prime}}_{k}=p_{k}\,\overline{P}_{k}$.

If we now divide $\overline{P\,^{\prime}}_{k}$ by the quantity of goods
$\overline{Q}_{k}$, we finally obtain the Price $\overline{P}^{\,*}_{k}$

$\overline{P}^{\,*}_{k}=b\,Ce^{-b\overline{Q}_{k}}\,.$ (36)

That is, the price $\overline{P}^{\,*}_{k}$, in reduced units um/umo, is
nothing more than the unit price determined statistically considering the
entire economic system. Note that if $\overline{Q}_{k}\to 0$, then the price
$\overline{P}^{\,*}_{k}\to b\,C$. In other words, when we study a single value
of $k$, the unit price is $b$. But, when all the values of $k$ are considered
statistically, then the unit price effective is the product $b\,C$. Observe
that the greater the amount of merchandise $\overline{Q}_{k}$ is traded, the
lower the price given the exponentially decreasing term in Eq. 36. In other
words, in a natural way, high prices are predicted when very few merchandises
are traded (what is known as the "retail price"), and lower prices when more
merchandise is traded or exchanged (”wholesale price“).

In short, what Eq. 36 tells us is that the price $\overline{P}^{\,*}_{k}$ of a
commodity or goods is exponentially decreasing with the quantity of goods
involved, as long as $b>0$ 151515In very rare cases, of academic value only,
where the unit price $b$ is a negative number, $\overline{P}^{\,*}_{k}$ is
also negative and exponentially increasing, provided that the Euler function
of degree $0$, $\overline{Q}_{k}$, is positive.. All of this without having
made any inference or assumption about the behavior of economic agents, except
that they randomly exchange merchandise among themselves.

### Price and Value

Chen (2005), in his The entropy theory of value, he made a similar proposal to
ours, but defining the value as $\mathcal{V}=\,-\,\log_{b}p_{k}$, relating it
to probability $p_{k}$, which he called “scarcity". That is, the greater the
“scarcity" (that is, the lower the probability), the greater the value
$\mathcal{V}$. In such a way that the value $\mathcal{V}(Q)$ of a certain
merchandise, is not more than the average of the value associated with each
probability. Concluding then Chen that

$\mathcal{V}(Q)=-\,\sum_{k=1}^{m}\,p_{k}\,log_{b}\,p_{k}\,.$ (37)

This proposal was made by Chen, observing exclusively the properties that
$\mathcal{V}$ had to fulfill: a) The value of two products must be higher than
the value of one of them. b) If two products are independent, that is, if the
products are not substitutes or partially substitutes for each other, then the
total value of the two products must be the sum of the two products. c) The
value of any product is nonnegative.

The only mathematical functions $f(x)$ that satisfy the three properties
mentioned above Applebaum (2008b) are of the form $f(x)=-\log_{b}x$. Note that
this function $f(x)$ is, except for the base of the logarithm, Eq. 27 for the
Euler function $\overline{S}^{\,*}_{k}$. The equivalent of the value function
$\mathcal{V}(Q)$ would be the statistical average of $\overline{S}^{\,*}_{k}$,
that is, the entropy $\overline{S}^{\,*}$.

##  11 Virtual dynamics

Through a computer program developed by us, we simulate this economic system
of $N$ agents who randomly exchange $Q$ goods, and which is equivalent to the
buying-selling process, where all $N$ agents can be both buyers as sellers. In
each step of the simulation, the only thing that the system registers is the
number of goods that each agent has.

For practical reasons, we set a unit of measure $\delta$, that is, the minimum
amount of merchandise, other than zero, that each agent can trade. Here we
chose as the unit of measure $\delta=1/20$. Thus we define
$\overline{Q}=\mathbf{\overline{Q}}/\delta$ as the Commodity Unit. Each of the
$N$ agents is assigned at the beginning of the dynamic the same amount of
merchandise, $Q/N$. In the merchandise exchange process, that is, in the
buying-selling process, at any step $t_{2}$, two randomly selected agents $i$
and $j$ exchange goods according to the following equations

$\overline{Q}_{i}(t_{2})=\mathsf{a}\,(\overline{Q}_{i}(t_{1})+\overline{Q}_{j}(t_{1}))\,,$
(38)
$\overline{Q}_{j}(t_{2})=(1-\mathsf{a})\,(\overline{Q}_{i}(t_{1})+\overline{Q}_{j}(t_{1}))\,.$
(39)

Where $\overline{Q}_{l}(t_{2})$ and $\overline{Q}_{l}(t_{2})$ are the goods
bought or sold by agent $l$ in step $t_{2}$; and $\overline{Q}_{l}(t_{1})$ and
$\overline{Q}_{l}(t_{1})$ are the goods bought or sold by agent $l$ in step
$t_{1}$, respectively. $t_{2}\,>\,t_{1}$ and $l=1$ or $2$. $\mathsf{a}$ is a
random number between $0$ and $1$. In this dynamic we will call a step when
the pair $i$,$j$ is chosen at random and they exchange goods, that is, they
buy-sell, according to the two previous equations. A run is completed when
$10$ million steps are totaled. Each result reported here is the average of
$30$ runs, unless otherwise noted. From the dynamics, the number of agents
$n_{k}$ who own a quantity of merchandise $\overline{Q}_{k}$ is statistically
determined.

The expressions 38 and 39 ensure that the goods sold for $i$ are exactly the
same as those obtained for $j$; in order to keep constant the total quantity
of goods within the system. It is clear that in any step of the dynamic, $i$
is the seller and $j$ is the buyer. But since agents $i$ and $j$ are chosen at
random in each step, $i$ could be the buyer and $j$ the seller in any other
step. We chose to work with $N=3000$ agents, which is an amount that
statistically represents reasonably well systems with many more agents. The
maximum amount of goods that any agent can access (“the roof" ),
$\mathbf{\overline{Q}}_{m}=\mathbf{Q}_{m}/N$, we made it vary between $10$
(Upper Limit  very high) and $1.1$ (Upper Limit  very low).

After obtaining the probabilities $p_{k}$, for each set ($\overline{Q}$,
$\overline{Q}_{m})$, $\overline{S}^{\,*}_{k}$ was graphed as a function of
$\overline{Q}_{k}$, such that assuming linear behavior for all $k$
161616Straight line $y(x)=m_{p}\,x+b_{o}$, where
$m_{p}=(y(x_{2})-y(x_{1}))/(x_{2}-x_{1})$ is the slope and $b_{o}$ is the
cutoff with $y(x)$., you have

$\overline{S}^{\,*}_{k}\cong b\,\overline{Q}_{k}-\overline{A}^{\,*}_{k}\,,$
(40)

thus allowing to determine, simultaneously, the slope of the straight line,
the unit price $b$, and the cut with the axis of the ordinates,
$\overline{A}^{\,*}_{k}$.

Figure 4: Probability $p_{k}$ vs $\overline{Q}_{k}$ , for different values of
the "roof" $\overline{Q}_{m}=\,$ $10$$(\bullet)$, $2$$(\blacksquare)$,
$1.5$$(\blacklozenge)$, $1.3$$(\blacktriangle)$, and $1.1$ $(\times)$.

##  12 Analysis of results

Without loss of generality, the goods and/or commodities that are exchanged
can be money itself. That money, in turn, could be the product of processes of
buying and selling merchandise other than money; or the money itself (measured
in $um$) involved in the purchase of foreign currency (measured in $um_{0}$).

Figure 5: Euler entropy $\overline{S}^{\,*}_{k}$ vs $\overline{Q}_{k}$, from
top to bottom, for three values of $\overline{Q}$ =
$6.67(\blacksquare),13.33(\blacktriangle)$ and $20(\bullet)$, and
$\overline{Q}_{m}=10$. The dashed lines ($-\,-\,-\,-\,-$) represent the linear
behavior of $\overline{S}^{\,*}_{k}$, Eq. 40.  Figure 6: Total Revenue
$\overline{P}$ vs Unit of Goods $\overline{Q}$, for positive values of
$\overline{P}$, and roofs $\overline{Q}_{m}=10$ $(\bullet)$, $2$
$(\blacksquare)$ and $1.5$ $(\blacklozenge)$. The segmented and dotted lines
are a guide for the eyes. The vertical lines indicate the maximum
$\overline{Q}_{MAX}$.  Figure 7: Total Revenue $\overline{P}$ vs Unit of Goods
$\overline{Q}$, for $\overline{Q}_{m}=10$ $(\bullet)$, $2$ $(\blacksquare)$
and $1.5$ $(\blacklozenge)$. The segmented and dotted lines are a guide for
the eyes. The vertical lines indicate where $\overline{P}$ changes sign. The
inset figure shows Marginal Revenue $P_{M}$ as a function of $\overline{Q}$
for $\overline{Q}_{m}=10$.  Figure 8:  Total Revenue $\overline{P}$, Entropy
$\overline{S}^{\,*}$, and $\overline{A}^{\,*}$, as a function of
$\overline{Q}$, for roofs $\overline{Q}_{m}=\,$ $10$ $(\bullet)$ and $2$
$(\blacksquare)$.  Figure 9:  The unit price $b$ as a function of
$\overline{Q}$, for different values of $\overline{Q}_{m}=\,$ $10$
$(\bullet)$, $2$$(\blacksquare)$, $1.5$$(\blacklozenge)$,
$1.3$$(\blacktriangle)$, and $1.1$ $(\times)$.  Figure 10:  Price
$\overline{P}^{\,*}_{k}$ as a function of $\overline{Q}_{k}$, for 4 values of
$\overline{Q}$ = $6.67$ up to $26.67$. The two inset figures show
$\overline{P}^{\,*}_{k}$ vs $\overline{Q}_{k}$, for $\overline{Q}$ = $73.33$
to $133.33$ (left figure), and $\overline{Q}$ = $140$ up to $160$ (right
figure). For all of them, $\overline{Q}_{m}=\,$ $10$. From top to bottom, in
all the figures, the value of $\overline{Q}$ increases.

Fig. 4 shows the probability profile $p_{k}$ as a function of the Goods Unit
$\overline{Q}_{k}$, with $\overline{Q}=20$ 171717In the work of Villagómez-
Manrique et al, $\mathbf{\overline{Q}}=1$., for different values of the
"ceiling" $\overline{Q}_{m}=10,\,2,\,1.5,\,1.3\,$ and $\,1.1$, in an economy
of agents that exchange money, of according to the dynamics 38 and 39. For all
purposes, a value of $\overline{Q}_{m}=10$ will be considered to be large
enough to be considered “without roof" .

The results presented here exactly reproduce those of Fig. 1 of the work by
Villagómez-Manrique and del Castillo-Mussot (2018). The most striking of these
probability profiles is the population inversion that is observed when
$\overline{Q}_{m}$ is reduced to values less than $2$, making it more likely
to observe more agents with large amounts of money and very few with little
money. For this to happen under these conditions, the price $b$ must be
negative. Villagómez-Manrique et al interpret it by saying that the absolute
temperature $T$ is negative. The equivalence between the equations 30 and 34
is evident.

As shown by the method used to compute $b$ and $\overline{A}^{\,*}_{k}$, the
Euler Entropy $\overline{S}^{\,*}_{k}$ vs $\overline{Q}_{k}$, according to Eq.
40, for three values of $\overline{Q}=6.67,\,13.33,\,20$, and a roof
$\overline{Q}_{m}=$10, Fig. 5. If $\overline{S}^{\,*}_{k}$ depended linearly
on $\overline{Q}_{k}$, over the entire range of values $\overline{Q}_{k}$,
then the result would be the straight line shown . It is clear that the
experimental results differ appreciably from this behavior, for high values of
merchandise $\overline{Q}_{k}$, even when the vast majority of agents maintain
low quantities of merchandise $\overline{Q}_{k}$ for positive unit prices. $b$
and $\overline{A}^{\,*}_{k}$ were determined, respectively, from the slope and
the intersection with the ordinates of the straight lines.

The Total Revenue $\overline{P}$ when buying-selling $\overline{Q}$ units of
goods, for three cases, when $\overline{Q}_{m}=$10, $2$ and $1.5$, is shown in
Fig. 6. The general shape of the $\overline{P}$ corresponds to what is
expected, Fig. 1. That is, as more goods are exchanged, the more the seller
gets, and the buyer pays, even though the unit price $b$ will be smaller and
smaller. Eventually $\overline{P}$ reaches a maximum, when the decrease in $b$
is balanced with the increase in $\overline{Q}$, after which, even when more
merchandise is traded, since the unit price is increasingly smaller, the total
quantity of money begins to decrease. The vertical lines show the commodity
values where the maximum values of $\overline{P}$, $\overline{Q}_{MAX}$ are
obtained 181818For the case of $\overline{Q}_{m}=2$ and $1.5$, the maximum of
the two coincide approximately, $\overline{Q}_{MAX}\cong 6.66$.. This general
behavior occurs for very high roofs, like $\overline{Q}_{m}=10$
($\overline{Q}_{MAX}\cong 26.66$), and those with much lower roofs, like
$\overline{Q}_{m}=$2 and $1.5$. For smaller values, not shown here, the
general behavior described above is still observed.

In the real world, the decrease of $\overline{P}$, after reaching the maximum,
is rarely observed 191919With the notable exception of the primary sector (raw
materials / natural resources: agriculture, mining, etc.)., since the economic
process ends up being unfeasible for the seller, and consequently it is
immediately stopped.

The complete graph of the behavior of $\overline{P}$ as a function of
$\overline{Q}$ is shown in Fig. 7. The vertical lines indicate when Total
Revenue $\overline{P}$ changes sign. In all the conditions studied this is
observed, but at very low roofs this change in sign appears at very low
commodity values. If the roof is high enough, as is the case with
$\overline{Q}_{m}=10$, the change in sign occurs at a fairly high value,
$\overline{Q}\cong 100$. Negative $\overline{P}$ values necessarily imply that
the unit price $b$ is also negative. For $\overline{Q}=2$ and $1.5$,
$\overline{P}$ changes sign by approximately $20$ and $15$, respectively.

If we now calculate the Marginal Revenue $P_{M}$ as the change in
$\overline{P}$ due to the increase in the quantity of goods $\overline{Q}$,
$P_{M}=\Delta{P}/\Delta{Q}=\Delta\overline{P}/\Delta\overline{Q}$, for the
case with roof $\overline{Q}_{m}=10$, the result is shown in the inset of Fig.
7. Qualitatively, it is what we would expect, Fig. 1, given the form of Total
Revenue, that is, as more merchandise is exchanged, the growth of what the
seller obtains "slows down" to the same extent, until it reaches zero, and
then begins to be negative, at which point the maximum of $\overline{P}$ is
exceeded, which represents, precisely, the maximum profit that the seller
obtains.

Total Revenue $\overline{P}\,=\,\overline{S}^{\,*}+\overline{A}^{\,*}$ has two
components, one entropic $\overline{S}^{\,*}$ and another “energetic"
$\overline{A}^{\,*}$. In Fig. 8 these three quantities are shown, for
$\overline{Q}_{m}=10$ and $2$. In both cases, the maximum of entropy
$\overline{S}^{\,*}$ and the change in concavity of $\overline{A}^{\,*}$,
coincide with the sign change in $\overline{P}$, and therefore also in the
unit price $b$.The increase in $\overline{P}$ (that is, when $P_{M}>0$) is the
commitment of two tendencies: the increase of the entropy of the system and
the decrease of the factor $\overline{A}^{\,*}$. When
$(\partial\overline{P}/\partial\overline{Q})=0$, then at the maximum
$(\partial\overline{S}^{\,*}/\partial\overline{Q})=-(\,\partial\overline{A}^{\,*}/\partial\overline{Q})$.
Thereafter, the decrease in $\overline{P}$ will be the consequence of a
greater decrease in $\overline{A}^{\,*}$ than the increase in
$\overline{S}^{\,*}$. As expected, the maximum of the entropy
$\overline{S}^{\,*}$ coincides with the point where $b$ and $\overline{P}$
change sign.

A crucial matter that we have investigated is the appearance of the price $b$
as a natural consequence of the way in which the merchandise is distributed.
Fig. 9 shows the monotonic decrease of $b$ as $\overline{Q}$ increases, for
four roofs $\overline{Q}_{m}=10,2,1.5,1.3$ and $1.1$. That is, as more
merchandise is traded, a natural decrease in the observed price is observed.
In other words, the behavior observed in real economic systems is evident
here, where the unit price decreases due to the increase in the quantity of
goods committed in the purchase-sale process. This is well-known Law of
demand. It is the result of the phenomenological observation that "when the
price of a good goes up, the quantity demanded goes down", the product of
certain fundamental assumptions of human behavior. For this reason, economists
"believe" that this is always, or almost always, true Landsburg (2011b).

The so-called retail price, that is, when buying-selling small quantities, is
statistically always higher than when buying-selling wholesale. All this
without individual agents acting rationally, looking for some kind of benefit.
Another way of interpreting it is that the more merchandise there is in the
system, the lower the observed unit price $b$ will be, and vice versa.

Keep in mind that in this work the independent variable is the amount of goods
$\overline{Q}$ or $\overline{Q}_{k}$, and that due to the distribution of
these among the agents, the price $b$ is generated naturally. From the
economy, on the contrary, given a unit price $b$ present in the economic
system, this determines the corresponding amount of merchandise traded
$\overline{Q}$ or $\overline{Q}_{k}$ by the agents .

On the other hand, note that at $\overline{Q}\cong 100$, $20$, and $5$, there
is a sign change in the respective unit price $b$, as expected. Likewise, in
the real world of economics, it is extremely rare to see negative prices, and
this has value here only for academic reasons. Clearly, in this case, the
density profiles $p_{k}$ must also present population inversion.

An advantage of the methodology described here is that it is possible to
answer, in a simple way, the reasons for a basic phenomenon observed in the
economy: The depreciation of the country’s currency due to the issuance of
money “created" by the Central Bank 202020We abstract from the secondary
expansion of the banks, as a consequence of the granting of credits, as
another source of “creation” of money., how much it injects money, so that the
government, for example, can meet salary commitments with its workers.

In our economic system model, whatever the roof is set, the price $b$
212121Remember that $b$ is a relative value of $um/um_{0}$. decreases as the
quantity of goods $\overline{Q}$ increases. If the commodity is money, then
money itself it depreciates when more money is in the hands of the economic
agents. However, the higher the roof $\overline{Q}_{m}$, the more “slowly"
will be its depreciation, Fig. 9. Below $\overline{Q}\cong 6.66$, indicated by
a vertical line, the behavior of the system is essentially the same,
regardless of the ceiling, noted in figure as region I. Between that line and
the second line located in $\overline{Q}\cong 26.66$, region II, Total Revenue
$\overline{P}$ is in the decreasing zone for all but the highest roof
$\overline{Q}_{m}=10$. From a practical point of view, this area is you’ll see
little in the real world for low roofs. Likewise, for above this line, region
III, $\overline{P}$ is also decreasing for $\overline{Q}_{m}=10$.

A decreasing $\overline{P}$, that is, a Marginal Revenue $P_{M}<0$, is not
convenient for whoever “sells" the money (the Central Bank), since it receives
less currency from the government for more money created. Here we have
considered the case when the Central Bank “sells" money ($um$) to the
government, who “pays" it with foreign currency ($um_{0}$) product of the
collection of taxes on foreign capital, or from the activity of its mixed or
state companies, which sell raw materials or manufactured products outside the
country. With this money, the government pays its national employees and
suppliers, recovering it later through the taxes it collects. Then it returns
to the Central Bank and it exchanges them for currencies, at a certain price,
which will be used to purchase products abroad, pay debt, etc., thus
completing the cycle.

So, one way to "slow down" the depreciation of money, and keep its price $b$
constant, due to the issuance of more money, is to raise the roof $Q_{m}$.
That is, the economic space created must be “expanded" so as to remain in the
area where the Marginal Revenue ${P}_{M}$ is always positive. This is what we
will call here indexing, that is, a mechanism tending to preserve the price of
money, that is, its purchasing power, by increasing the flow of money in the
economy. The dotted horizontal line in Fig. 9 allows us to understand this
process, focusing mainly on region II. It is clear that we must "jump" to
higher and higher ceilings if we want to keep the price $b$ of money stable,
as we increase its quantity in the economy.

An issue that may also be of interest is to estimate the value of the price at
each site $k$, $\overline{P}^{\,*}_{k}$. As can be seen in Fig. 10,
$\overline{P}^{\,*}_{k}$ follows the general decreasing behavior of $b$, as
the quantity of merchandise increases $\overline{Q}$. In other words, this is
the “Law of demand" before described. The sign of $\overline{P}^{\,*}_{k}$
also corresponds to the sign of $b$. The unit price $b$, with
$\overline{Q}_{m}=10$, is positive for $0\,\leq\,\overline{Q}\,\leq\,100$, and
negative for $\overline{Q}\,>\,100$.

##  13 Conclusions

In this work we have developed a methodology based on thermodynamic and
mechanical-statistical concepts, which facilitate the conceptual understanding
of phenomena that have been widely observed in real economic systems, such as
the price, the Law of demand, the Total Revenue, and the depreciation of a
currency. The latter occurs when the economic system is flooded with more
money, and the way to index prices is by expanding the economy, that is,
increasing the "roof" of the economic system. These phenomena arise as a
consequence of two quantities that compete with each other: configurational
entropy and a term such as "free energy", a product of the way in which goods
are distributed between the economic agents. The unit price $b$, whose value
emerges naturally, is equivalent to $\beta=1/k_{B}T$ in classical
thermodynamic systems, presenting situations where there are negative prices.
In short, using Euler’s theorem of homogeneous functions has made it possible
to connect economic properties and phenomena with their thermodynamic
equivalents.

## Acknowledgments

We would like to thank Marcelo del Castillo-Mussot for his valuable opinions
and recommendations.

## References

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