A Four Laws Structure for Looking at Economics through the Eyes of Thermodynamics

Abstract

Thermodynamics is based on its Four Laws structure. Based on Thermodynamics and its Four Laws structure, this work presents and explores its economic counterpart, proposing an economic Four Laws structure to look at Economics through the eyes of Thermodynamics. The Economic Zeroth Law is based on the merchandise economic thermal equilibrium, thus defining the economic temperature. The Economic First Law is based on balance equations for the units of merchandise and monetary species. The economic Second Law is based on the observation that the traded merchandise units are transferred in the increasing unit price direction (in the decreasing economic temperature direction). This allows economic entropy to be defined through a differential equation, as well as the definition of economic irreversibility and economic entropy generation. The economic Third Law states that the merchandise economic entropy of an economic system at the null economic temperature (at an infinite merchandise unit price) is null. Conjugation of the economic Second and Third Laws sets how to evaluate the absolute value of the economic entropy of an economic system. The presented developments setting the proposed economic Four Laws structure introduce and explore a large set of concepts, variables, and relations that are of major relevance for looking at Economics through the eyes of Thermodynamics.

1. Introduction

Thermodynamics is well established through its concepts, variables, properties, and the Four Laws structure [1,2,3]. The same happens with Economics, which is well established through its concepts, variables, Laws, and structure [4,5]. New interpretations of well-established domains are always interesting and important, even when using analogies. They allow for different looks, insights, interpretations, analyses, and diagnoses. Such an approach promotes cross-fertilization between different areas, allowing new interpretations and developments in different fields that are otherwise closed. This is also a scientific and pedagogical opportunity, to open new avenues for exploring the fields involved. From a pedagogical viewpoint, it allows alternative teaching and learning approaches, the more easily understandable of one helping to understand the other. Even new teaching/learning methodologies could emerge from the similarities between the different fields, always taking to mind that care is needed when teaching/learning using analogies. If this is true in general, it also applies to Thermodynamics and Economics.

This work proposes a Four Laws structure, analogous to the Thermodynamics Four Laws structure, to look at Economics through the eyes of Thermodynamics. In addition to the proposed Four Laws structure, the presented developments introduce and explore a large set of concepts, variables, and relations that are of major relevance for the same purpose of looking at Economics through the eyes of Thermodynamics.

Research on Thermodynamics and Economics analogies is pointed out in [6] and the references therein. Many of these studies are based on the original works of Georgescu-Roegen [7,8,9]. In [7], he relates Mechanics, Thermodynamics, production, and Economics, focusing on the material entropy law, economic activity, and sustainable evolution. In [8], he dissertates on evolution, entropy, order, probability, cause, purpose, value, and development. In [9], he explores connections between energy, entropy, Economics, evolution, ecology, and ethics, considering the Second Law’s one-way direction. Georgescu-Roegen’s [7,8,9] ideas and works are grounded in economic scarcity and long-term sustainability (the material entropy increase precludes the possibility of full recycling). These are theoretical treatises far from the starting points, objectives, and purposes of the approach proposed in [6] and followed in the present work. Later literature [10] claimed that Georgescu-Roegen’s concept of material entropy is incorrect. It is noted, however, that the works of Georgescu-Roegen are still the yeast for studies relating Economics and Thermodynamics, as recently condensed in [11].

Works in References [12,13,14,15] are examples of studies attempting to set analogies between Economics and Thermodynamics, relating variables and parameters (mainly in the form of state equations and equilibrium conditions) describing economic systems, similarly as made in Thermodynamics [1,2,3]. Quevedo and Quevedo [12] propose a statistical thermodynamic treatment of economic systems. Belabes [13] dissertates on how Thermodynamics can be useful to economists and energy engineers. Rashkovskiy [14,15] describes economic systems and processes based on an analogy between the parameters of thermodynamic and economic systems (taken as markets), setting the economic analogs of internal energy and temperature. Saslow [16] explores the economic analogs of energy, entropy, equilibrium, thermometry, and statistical mechanics. However, he does not consider the one-way character of the Second Law or the dynamic character of economic processes. In recent work, Roddier [17] used the Laws of Thermodynamics to try to explain economic behaviors. Previously referred works are, however, far from the starting points, objectives, purposes, and results of the work in [6] and the present work.

Thermodynamic systems are assemblies of statistically well-behaved large numbers of particles, allowing their description by macroscopic properties like temperature and pressure. State equations, which can be explicitly known or implicitly assumed, impose relationships between the macroscopic properties [1,2,3]. In Economics, statistical models do not apply when small numbers of goods, services, and money are under analysis. Additionally, arbitrary decisions need to be accommodated by the adopted economic models [6]. The present work complements the work in [6]. Their main relevance is that they differ from the starting points, objectives, purposes, and results of the works in the literature, setting analogies between Thermodynamics and Economics. They are based on the non-equilibrium dynamics and financial value generation in merchandise trading. Merchandise transfers in trading operations, traded merchandise, economic irreversibility, and merchandise economic entropy generation (merchandise financial value generation) are some of their strongest words. Their main equations are not state equations, equilibrium conditions, or relations between properties but accounting and balance equations, which apply equally to small and large numbers of goods, services, and money in both equilibrium and non-equilibrium processes. The starting points, developments, results, and conclusions make them different from other works in the literature trying to set analogies between Economics and Thermodynamics, as explained in more detail in [6].

Work in [6] is based on the economic analogs of the First and Second Law of Thermodynamics. However, it does not refer to the Zeroth and Third Laws, and the Second Law is assumed to be known and already set. Analyses, results, discussions, and reflections in [6] are based on the economic analogs of the First and Second Laws. However, for a more complete look at Economics through the eyes of Thermodynamics, the Zeroth and Third Laws must be introduced, not only referring to them but also including the developments that led to them. The same happens with the economic analog of the Second Law, whose introduction and setting require a series of developments, starting from base observations, as made for the Second Law of Thermodynamics [3]. These are the reasons and objectives of the present paper: to introduce the economic analogs of the Zeroth and Third Laws, including the developments leading to them and the developments that, starting from the base observations, lead to the setting of the economic analog of the Second Law. Such developments allow for the definition of economic entropy and economic entropy generation. Knowing the developments leading to the setting of the Four Laws deepens our knowledge and understanding of these Laws. Compared to the work in [6], no significant developments have been made concerning the economic analog of the First Law. However, to set up a unified Four Laws structure, the economic analogs of the Four Laws are introduced and worked together in the present work. Additionally, it must be mentioned that to the best of the author’s knowledge, such a unified Four Laws structure has not been proposed before.

This work sets and proposes a Four Laws structure analogous to the Thermodynamics Four Laws structure for looking at Economics through the eyes of Thermodynamics, which is based on the following main economic principles:

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If the two economic systems remain in economic contact (possibility of traded merchandise exchanges) for a sufficiently long time, an equilibrium situation is reached so that the traded merchandise transfers between them cease. These economic systems are in economic thermal equilibrium and are at the same economic temperature. Economic thermal equilibrium allows for the definition of economic temperature. This is the basis of the Economic Zeroth Law, set and introduced in Section 2.

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The economic systems include merchandise and monetary units, which can be exchanged. The merchandise and monetary units in the system, the merchandise and monetary units exchanged by the system, and the merchandise and monetary units generation in the system obey the balance equations. In this case, the units balance equations. This is the Economic First Law, set and introduced in Section 3.

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Traded merchandise units are transferred in the direction of increasing unit prices (in the direction of decreasing economic temperature). This is the base observation of the economic Second Law, set and introduced in Section 4. Developments made from this base observation allow the economic entropy to be defined using a differential equation. Economic irreversibility and entropy generation are also introduced and discussed in Section 4. Economic cycles, analogous to thermodynamic cycles, are crucial for this purpose. The developments in Section 4 are of major relevance for looking at Economics through the eyes of Thermodynamics.

-

When the economic temperature of the system is decreased to zero, its merchandise economic entropy decreases to zero. This is the economic Third Law, set and introduced in Section 5.

The work in [6] assumes that the economic First and Second Laws are already set, and they are used to look at Economics through the eyes of Thermodynamics. It does not consider, however, the Economic Zeroth Law, the rich and lengthy developments from the base economic observations up to the definition of economic entropy and of the merchandise economic entropy generation. Additionally, it does not consider the economic Third Law.

The main concepts, definitions, variables, and equations involved in the developments leading to the proposed Four Laws structure for looking at Economics through the eyes of Thermodynamics include, by order of appearance:

In Section 2:

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Economic thermal equilibrium,

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Economic temperature,

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Economic Zeroth Law.

In Section 3:

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Traded merchandise,

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Merchandise transfer interactions,

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Merchandise wealth,

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Merchandise units generation,

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Merchandise reservoir,

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Monetary transfer interactions,

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Monetary wealth,

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Monetary units generation,

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Monetary exchanges in trading operations,

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Units balance equations,

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Economic First Law.

In Section 4:

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Economic Second Law,

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Economic Kelvin-Planck statement of the Second Law,

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Economic Clausius statement of the Second Law,

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Economic Carnot cycle,

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Economic reversibility, and economic irreversibility,

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Absolute economic temperature scale and absolute economic unit price scale,

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Economic Bucher diagram,

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Economic Clausius inequality,

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Economic entropy definition through a differential equation,

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Economic entropy generation,

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Economic entropy transfer interaction,

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Economic entropy balance equation,

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Merchandise economic entropy generation (profit generation) in merchandise transfer through a finite economic temperature difference,

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A new insight into the differential equation defining the merchandise economic entropy.

In Section 5:

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Absolute value of the merchandise economic entropy,

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Economic Third Law.

2. The Economic Zeroth Law

Developments in Thermodynamics leading to its Four Laws structure are based on temperature and heat, which flow spontaneously in the direction of decreasing temperature. These developments start with the thermal equilibrium concept and empiric temperature (measured by a thermometer). The Zeroth Law of Thermodynamics has its roots in the experimental observations of thermal equilibrium [1,2,3], two systems in thermal equilibrium having the same temperature value.

Similarly, developments leading to the proposed Four Laws structure are based on economic temperature and traded merchandise, which flows spontaneously in the economic temperature-decreasing direction. This section introduces the concept of economic thermal equilibrium and the economic analog of the empiric temperature, referred to as the empiric economic temperature. It is the inverse of the unit price of the traded merchandise (a good or service). The Economic Zeroth Law is set based on the concept of economic thermal equilibrium, which states that two economic systems in economic thermal equilibrium have the same economic temperature value.

2.1. Concepts and Definitions

The contents of this section closely follow some parts of [6].

The economic system is delimited by a boundary, an imaginary frontier separating what is under analysis from what is out of analysis. Every economic system is open, as its boundaries can be crossed by material (tangible, hardware) or immaterial (intangible, software) goods, services, and money.

An economic system can be composed of $N_M$ [U] merchandise units (of goods or services), and $N_m$ [U] monetary units. It is not usual to attribute units when counting things, but for an effective analogy between Thermodynamics and Economics, the unit ‘Unit’ is used and denoted by [U]. As many new ideas, concepts, variables, and relations appear in this work for the second time, the first one being in [6], units are indicated inside square brackets for accuracy and easier reading and understanding.

The economic temperature, $T_E$ [U/€], of merchandise (good or service) is defined as the inverse of its unit price $p$ [€/U], that is,

$$T_E={\displaystyle \frac{1}{p}}[\mathrm{U}/€]$$

(1)

Euro [€] is used as the monetary unit. However, other monetary units can be considered similarly. Traded merchandise flows spontaneously in the increasing unit price direction; that is, given Equation (1), traded merchandise flows spontaneously in the decreasing economic temperature direction. In the present context, spontaneous means occur under normal conditions during normal trading operations.

Temperature is a property of a thermodynamic system [1,2,3]. Different chemical species in a thermodynamic system share the same temperature range [1,2,3]. The proposed economic temperature is not a property of the economic system but a property of one traded merchandise, as it depends on the context, which influences its unit price when crossing the economic system’s boundary. Different merchandise species in the same economic system experience different economic temperatures. A more complete and accurate definition of the economic temperature must refer to merchandise species i, $T_{E,M,i}=1/p_i$ [U/€]. However, once this is made clear, for conciseness, only economic temperature $T_E$ [U/€] will be used whenever possible if it does not result in confusion.

If an economic system is composed of different merchandise species, each with its unit price and thus with its economic temperature, analysis must be conducted considering each merchandise species separately and then adding the contributions of all merchandise species.

2.2. The Economic Zeroth Law

The Zeroth Law of Thermodynamics is based on the concept of thermal equilibrium, and temperature is a thermodynamic property whose value is the same for systems in thermal equilibrium.

Similarly, the Economic Zeroth Law is based on the economic thermal equilibrium concept, and the economic temperature value is the same for economic systems in economic thermal equilibrium. For simplicity, it is assumed that the economic systems considered are composed of units of one and the same merchandise species only.

If two economic systems, A and B, are able to receive or release traded merchandise units and stay in economic contact for a sufficiently long time, a situation is reached in which they cease to receive or release traded merchandise units. In this situation, economic systems A and B are said to be in economic thermal equilibrium; they no longer receive or release traded merchandise units unless some conditions change, and they have the same economic temperature value (and given Equation (1), they have the same merchandise unit price value). There is no motivation to continue trading merchandise if the unit prices (if the economic temperatures) of merchandise units in these economic systems are equal.

The analysis can be expanded by considering the three economic systems, A, B, and C, as illustrated in Figure 1. If system A is in economic thermal equilibrium with system C, then they have the same economic temperature value. If system B is in economic thermal equilibrium with system C, they have the same economic temperature value. This means that systems A and B are in economic thermal equilibrium. In such an economic thermal equilibrium situation,

$$\begin{array}{l}{T}_{E,A}=T_{E,C}\\ T_{E,B}=T_{E,C}\end{array}\}{T}_{E,A}=T_{E,B}[\mathrm{U}/€]\iff \begin{array}{l}{p}_A=p_C\\ p_B=p_C\end{array}\}{p}_A=p_B[€/\mathrm{U}]$$

(2)

This is the mathematical statement of the Economic Zeroth Law.

The concept of thermal equilibrium allows defining the property empiric temperature (measured with a thermometer), a property whose numerical value is the same for systems in thermal equilibrium. The same happens with the economic thermal equilibrium concept, which allows defining the property of economic temperature, the property whose numerical value is the same for economic systems in economic thermal equilibrium. From a formal viewpoint, the left-hand side of Equation (2) must appear before Equation (1), and only after that the right-hand side of Equation (2). However, as unit price is a familiar concept, contrary to what happens with the economic thermal equilibrium and economic temperature concepts, Equations (1) and (2) appear in the text in that order for readability and understanding, even if it is not the right one from the formal viewpoint.

An economic temperature difference (a unit price difference) can be seen as the driving force for traded merchandise transfer, and after this driving force vanishes (economic thermal equilibrium conditions), no more merchandise is transferred in normal trading operations (equal merchandise unit prices for the potential merchandise seller and the potential merchandise buyer). It will be seen in Section 4 that the true driving force for merchandise transfer in trading operations is the economic entropy generation (financial value generation) and not exactly the economic temperature difference or the unit price difference.

In this sense, a monetary transfer corresponds to an economic thermal equilibrium process, which is no more than an exchange process because there is no change in the unit price (no change in the economic temperature) of the exchanged monetary units in such a process.

3. The Economic First Law and the Merchandise and Monetary Units Balance Equations

In Thermodynamics, once the Zeroth Law is set, the next step is to define the mass and energy and the different forms of energy. The First Law of Thermodynamics, also referred to as the Energy Conservation Principle, can be introduced in many ways, the most common one in Engineering Thermodynamics being through the energy balance equation [1,2,3]. This is also the most adequate form for the present purposes.

This section begins by defining the merchandise and monetary units to be accounted for in an economic system, in a similar way to that used for energy and forms of energy in Thermodynamics. This includes defining the traded merchandise units involved in trading operations and the wealthy merchandise units, these not involved in trading operations. An economic system may also be composed of monetary units that may be exchanged in trading operations or set aside as monetary wealth.

Accounting for all detained, exchanged, and generated merchandise and monetary units leads to the units’ balance equation. The total number of units in the balance equation sets the Economic First Law.

The following contents of this section closely follow those in Section 2 of [6].

3.1. Concepts and Definitions

3.1.1. Merchandise and Monetary Units in the Economic System

The number of merchandise units in the economic system, composed of $N_{M,i}$ [U] merchandise units of different species i, is

$$N_M={\displaystyle \sum _i{N}_{M,i}}[\mathrm{U}]$$

(3)

and the number of monetary units in the economic system, composed also of $N_{m,k}$ [U] monetary units of different monetary species k, is

$$N_m={\displaystyle \sum _k{N}_{m,k}}[\mathrm{U}]$$

(4)

The total number of units in the economic system is the sum of the number of units of its (merchandise and monetary) constituents

$$N=N_M+N_m={\displaystyle \sum _i{N}_{M,i}}+{\displaystyle \sum _k{N}_{m,k}}[\mathrm{U}]$$

(5)

3.1.2. Financial Value of the Merchandise and Monetary Units in the Economic System

The contribution of the merchandise species to the financial value of the economic system is the sum, extended to all the merchandise species in the system, of the products of the merchandise units of species i, $N_{M,i}$ [U], by its unit price, $p_i$ [€/U],

$$V_{F,M}={\displaystyle \sum _i{N}_{M,i}·p_i}[€]$$

(6)

The contribution of the monetary species to the financial value of the economic system is the sum, extended to all the monetary species in the system, of the products of the monetary units of species k, $N_{m,k}$ [U], by its unit financial value, $F_k$ [€/U],

$$V_{F,m}={\displaystyle \sum _k{N}_{m,k}·F_k}[€]$$

(7)

where $F_k$ [€/U] is the exchange rate from the monetary species k to Euro.

The financial value of the economic system, composed of merchandise and monetary units as given by Equation (5), is thus

$$V_F=V_{F,M}+V_{F,m}={\displaystyle \sum _i{N}_{M,i}·p_i}+{\displaystyle \sum _k{N}_{m,k}·F_k}[\mathrm{U}]$$

(8)

The financial value of the economic system is influenced by the abundance or scarcity of the merchandise it contains in its neighboring and by the context in which it is considered.

3.2. Merchandise and Monetary Transfer Interactions

The merchandise and monetary units exchanged by the economic system are referred to as unit transfer interactions. The analysis considers the merchandise and monetary unit transfer interactions crossing the system’s boundary, and what happens (accumulation and generation) with the merchandise and monetary units in the system (inside the boundary).

A dot over a variable indicates the flow rate of that variable. As illustrated in Figure 2, an economic system can exchange merchandise and monetary units through the system’s boundary as:

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Traded merchandise, $\dot{M}$ [U/s]: number of merchandise units in transit per unit of time (which may be composed of traded merchandise flow rates ${\dot{M}}_i$ [U/s] of different merchandise species i), driven by a potential (economic temperature difference), $\dot{M}$ [U/s] flowing spontaneously in the decreasing economic temperature direction (in the increasing unit price direction). These are merchandise units that are involved in trading operations.

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Merchandise wealth, ${\dot{W}}_M$ [U/s]: number of merchandise units in transit per unit of time (which may be composed of merchandise wealth flow rates ${\dot{W}}_{M,i}$ [U/s] of different merchandise species i), not driven by any potential, being thus not transferred by trading reasons. These are merchandise units that are not in the market.

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Monetary wealth, ${\dot{W}}_m$ [U/s]: number of monetary units in transit per unit of time (which may be composed of monetary wealth flow rates ${\dot{W}}_{m,k}$ [U/s] of different monetary species k), not driven by any potential, being thus not transferred by trading reasons. These are monetary units that are not used in trading operations but are set aside as monetary wealth.

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Monetary transfer, $\dot{m}$ [U/s]: number of monetary units in transit per unit of time (which may be composed of monetary flow rates ${\dot{m}}_k$ [U/s] of different monetary species k), not driven by any potential, being thus not transferred by trading reasons. They are in the market, used in trading operations, and are used for exchanges by traded merchandise units.

Traded merchandise and merchandise wealth can be either material (hardware) or immaterial (software). The monetary flow rates mean monetary units in transit. They are considered non-traded because their unit prices do not change when they are exchanged in trading operations.

The (merchandise and monetary) wealth transfer interactions are the analogs of the mechanical work transfer interactions in Thermodynamics. The traded merchandise and monetary exchange transfer interactions are the analogs of the heat transfer interactions in Thermodynamics.

3.3. The Units Balance Equations

3.3.1. General Form of the Units Balance Equation

As made for the (total) energy balance equation [1,2,3], an equation can be similarly written by setting the balance of the total number of units in an economic system. Written in the same form as usual in Engineering Thermodynamics [1,2,3], such an equation sets that

$$\begin{array}{ll}\underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{\left(\begin{array}{l}\mathrm{time}\mathrm{rate}\mathrm{of}\\ \mathrm{change}\mathrm{of}\mathrm{the}\\ \mathrm{number}\mathrm{of}\mathrm{units}\\ \mathrm{in}\mathrm{the}\mathrm{system}\end{array}\right)}}=& \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{net}\mathrm{flow}\mathrm{rate}\mathrm{of}\\ \mathrm{traded}\mathrm{merchandise}\\ \mathrm{units}\mathrm{entering}\mathrm{the}\\ \mathrm{system}\end{array}\right)-\left(\begin{array}{l}\mathrm{net}\mathrm{flow}\mathrm{rate}\mathrm{of}\\ \mathrm{wealth}\left(\mathrm{non}\text{-}\mathrm{traded}\right)\\ \mathrm{merchandise}\mathrm{units}\\ \mathrm{leaving}\mathrm{the}\mathrm{system}\end{array}\right)}}-\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{net}\mathrm{flow}\mathrm{rate}\mathrm{of}\\ \mathrm{wealth}\mathrm{monetary}\\ \mathrm{units}\mathrm{leaving}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}+\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{flow}\mathrm{rate}\mathrm{of}\\ \mathrm{exchanged}\mathrm{monetary}\\ \mathrm{units}\mathrm{entering}\mathrm{the}\\ \mathrm{system}\end{array}\right)-\left(\begin{array}{l}\mathrm{flow}\mathrm{rate}\mathrm{of}\\ \mathrm{exchanged}\mathrm{monetary}\\ \mathrm{units}\mathrm{leaving}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}+\\ & \underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{\left(\begin{array}{l}\mathrm{net}\mathrm{generation}\\ \mathrm{rate}\mathrm{of}\mathrm{merchandise}\\ \mathrm{units}\mathrm{in}\mathrm{the}\\ \mathrm{system}\end{array}\right)+\left(\begin{array}{l}\mathrm{net}\mathrm{generation}\\ \mathrm{rate}\mathrm{of}\mathrm{monetary}\\ \mathrm{units}\mathrm{in}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}\end{array}[\mathrm{U}/\mathrm{s}]$$

3.3.2. Units Balance Equation for All the Merchandise and Monetary Units

The units balance equation for all the merchandise and monetary units sets that the time rate of change of the number of units of all merchandise and monetary species in the economic system equals the sum of the flow rates ${\dot{M}}_{i,j}$ [U/s] of all traded merchandise species i exchanged by the economic system with the $N_M$ merchandise reservoirs, minus the merchandise wealth rates ${\dot{W}}_{M,i}$ [U/s] of all merchandise species i exchanged by the economic system, minus the monetary wealth rates ${\dot{W}}_{m,k}$ [U/s] of all monetary species k exchanged by the economic system, plus the monetary rates ${\dot{m}}_k$ [U/s] of all the monetary species k entering the economic system, minus the monetary rates ${\dot{m}}_k$ [U/s] of all the monetary species k leaving the economic system, plus the generation rates ${\dot{N}}_{M,g,i}$ [U/s] and ${\dot{N}}_{m,g,k}$ [U/s] of, respectively, all merchandise and monetary species in the economic system, as

$$\begin{array}{ll}\underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{{\displaystyle \sum _i\left(\frac{d{N}_{M,i}}{dt}\right)}+{\displaystyle \sum _k\left(\frac{d{N}_{m,k}}{dt}\right)}}}=& \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{{\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}{\dot{M}}_{i,j}}\right)}-{\displaystyle \sum _i{\dot{W}}_{M,i}}-{\displaystyle \sum _k{\dot{W}}_{m,k}}}}+\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{{\displaystyle \sum _k\left({\displaystyle \sum _{in}{\dot{m}}_k}\right)}\hspace{0.17em}-{\displaystyle \sum _k\left({\displaystyle \sum _{out}{\dot{m}}_k}\right)}}}+\\ & \underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{{\displaystyle \sum _i{\dot{N}}_{M,g,i}}+{\displaystyle \sum _k{\dot{N}}_{m,g,k}}}}\end{array}[\mathrm{U}/\mathrm{s}]$$

(9)

which is graphically illustrated in Figure 2. The (merchandise and monetary) wealth transfer interactions ${\dot{W}}_{M,i}$ [U/s] and ${\dot{W}}_{m,k}$ [U/s] are taken as positive when released by the economic system, assuming that the positive effect of an economic engine is the delivered merchandise wealth, similar to Thermodynamics, where the delivered mechanical work is the positive effect of a thermal engine. The traded merchandise transfer interactions ${\dot{M}}_{i,j}$ [U/s] are positive when entering the economic system and negative otherwise.

The concept of a merchandise reservoir is defined similarly to a heat reservoir in Thermodynamics, whose temperature remains unchanged regardless of the heat it receives or releases. Thus, a merchandise reservoir is an economic system whose economic temperature remains unchanged and is independent of the traded merchandise units it receives or releases.

3.3.3. Separated Balance Equations for All the Merchandise and Monetary Units

${\dot{M}}_j={\displaystyle \sum _i{\dot{M}}_{i,j}}$ [U/s] can be decomposed into its components ${\dot{M}}_{i,j}$ [U/s] associated with each merchandise species i. Similarly, ${\dot{W}}_M={\displaystyle \sum _i{\dot{W}}_{M,i}}$ [U/s] can be decomposed into its components ${\dot{W}}_{M,i}$ [U/s] associated with each merchandise species i. The same happens also with the generation rates of merchandise units in the economic system, being ${\dot{N}}_{M,g}={\displaystyle \sum _i{\dot{N}}_{M,g,i}}$ [U/s]. By its turn, and similarly, ${\dot{W}}_m={\displaystyle \sum _k{\dot{W}}_{m,k}}$ [U/s] can be decomposed in its components ${\dot{W}}_{m,k}$ [U/s] associated with each monetary species k. The same also happens with the generation rates of monetary units in the economic system, being ${\dot{N}}_{m,g}={\displaystyle \sum _k{\dot{N}}_{m,g,k}}$ [U/s]. All previous decompositions are possible, assuming that the values of the referred variables for each merchandise species i and monetary species k are not coupled with the values of the corresponding variables for any other merchandise or monetary species.

Equation (9) can be seen as resulting from two balance equations, one for all the merchandise species

$${\displaystyle \sum _i\left(\frac{d{N}_{M,i}}{dt}\right)}={\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}{\dot{M}}_{i,j}}\right)}-{\displaystyle \sum _i{\dot{W}}_{M,i}}+{\displaystyle \sum _i{\dot{N}}_{M,g,i}}[\mathrm{U}/\mathrm{s}]$$

(10)

and one for all the monetary species

$${\displaystyle \sum _k\left(\frac{d{N}_{m,k}}{dt}\right)}=-{\displaystyle \sum _k{\dot{W}}_{m,k}}+{\displaystyle \sum _k\left({\displaystyle \sum _{in}{\dot{m}}_k}\right)}\hspace{0.17em}-{\displaystyle \sum _k\left({\displaystyle \sum _{out}{\dot{m}}_k}\right)}+{\displaystyle \sum _k{\dot{N}}_{m,g,k}}[\mathrm{U}/\mathrm{s}]$$

(11)

This can be made because there are no conversions of merchandise units into monetary units or conversions of monetary units into merchandise units. What happens in trading operations are exchanges of merchandise units by monetary units and of monetary units by merchandise units.

3.3.4. Relating the Merchandise and Monetary Transfer Interactions in Trading Operations

Even if Equations (10) and (11) can be taken separately, they are coupled in trading operations, as it is implied that the purchase (income into the economic system) of $M_p$ [U] traded merchandise units at the unit price $p_p$ [€/U] is associated with the outcome of the monetary financial value $\left|m_p\right|\times 1$ [€], obeying $M_p·p_p=\left|m_p\right|\times 1$ [€], and that the selling (outcome from the economic system) of $\left|M_s\right|$ [U] traded merchandise units at the unit price $p_s$ [€/U] is associated with the income of the monetary financial value $m_s\times 1$ [€], obeying $\left|M_s\right|·p_s=m_s\times 1$ [€], as illustrated in Figure 3. It is through relations of the type

$$\{\begin{array}{l}{M}_p·p_p=\left|m_p\right|\times 1\\ \left|M_s\right|·p_s=m_s\times 1\end{array}[€]$$

(12)

that Equations (10) and (11) are coupled in trading operations.

3.3.5. Units Balance Equation for a Merchandise Species and for a Monetary Species

Looking at balance Equations (10) and (11), they may themselves be decomposed into a set of balance equations, one for each merchandise species i

$${\displaystyle \frac{d{N}_{M,i}}{dt}}={\displaystyle \sum _{j=1}^{N_M}{\dot{M}}_{i,j}}-{\dot{W}}_{M,i}+{\dot{N}}_{M,g,i}[\mathrm{U}/\mathrm{s}]$$

(13)

and one for each monetary species k

$${\displaystyle \frac{d{N}_{m,k}}{dt}}=-{\dot{W}}_{m,k}+{\displaystyle \sum _{in}{\dot{m}}_k}\hspace{0.17em}-{\displaystyle \sum _{out}{\dot{m}}_k}+{\dot{N}}_{m,g,k}[\mathrm{U}/\mathrm{s}]$$

(14)

3.3.6. Generation of Merchandise and Monetary Units in the Economic System

Contrarily to what happens with energy, which is conserved [1,2,3], units of merchandise species and monetary species can be created/generated or destroyed in the economic system. For example, new ideas, new knowledge, or new combinations of goods or services originate new merchandise units without forcing the destruction of other merchandise units for that, cases for which ${\dot{N}}_{M,g,i}\ge 0$ [U/s]. However, situations can exist in which units of merchandise are destroyed in the economic system, cases for which ${\dot{N}}_{M,g,i}\le 0$ [U/s]. For example, glasses can be broken (destroyed) in an economic system without implying the generation of other merchandise units. New ideas that can be taken as merchandise wealth or placed in the market as traded merchandise, giving them unit prices, can be generated without implying the destruction of other merchandise units in the economic system.

The generation of monetary units may occur only in the central banks and in the common economic operations ${\dot{N}}_{m,g,k}=0$ [U/s].

3.3.7. Notes on the Units’ Balance Equations

The merchandise wealth rate ${\dot{W}}_{M,i}$ [U/s] in Equations (9), (10) and (13) is the sum of negative (received) and positive (released) i merchandise wealth rates. Similarly, the monetary wealth rate ${\dot{W}}_{m,k}$ [U/s] in Equations (9), (11) and (14) is the sum of negative (received) and positive (released) k monetary wealth rates.

The units of all merchandise species i transferred in trading operations with the $N_M$ merchandise reservoirs are considered in the term ${\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}{\dot{M}}_{i,j}}\right)}$ [U/s]. The merchandise transfer rates can be motivated by trading (${\dot{M}}_{i,j}$ [U/s]) or not (${\dot{W}}_{M,i}$ [U/s]), as considered separately in Equations (9), (10) and (13). All the traded merchandise transfer rates ${\dot{M}}_{i,j}$ [U/s] motivated by trading and all the merchandise wealth rates ${\dot{W}}_{M,i}$ [U/s], these not motivated by trading, are included respectively in the first and second terms on the right-hand sides of Equations (9), (10) and (13).

The monetary wealth rates ${\dot{W}}_{m,k}$ [U/s] transferred can have material (tangible, as coins or monetary billets) or immaterial (intangible, like cheques or electronic transfers) components. The monetary units exchanged by the economic system are considered separately in Equations (9), (11) and (14) through the terms ${\displaystyle \sum _k\left({\displaystyle \sum _{in}{\dot{m}}_k}\right)}\hspace{0.17em}-{\displaystyle \sum _k\left({\displaystyle \sum _{out}{\dot{m}}_k}\right)}$, as their transfers are not motivated by trading (they are not transferred with profit generation purposes, but just exchanged by traded merchandise in the trading operations). Traded merchandise transfers occur from lower unit prices to higher unit prices (from higher economic temperatures to lower economic temperatures), with profit generation (financial value generation) purposes.

3.3.8. Differential Form of the Units Balance Equations

Multiplying Equation (9) by dt allows obtaining the differential form of the total units balance equation for an infinitesimal economic process as

$$\begin{array}{ll}{\displaystyle \sum _{i}d{N}_{M,i}}+{\displaystyle \sum _{k}d{N}_{m,k}}=& {\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\delta M_{i,j}}\right)}-\\ & {\displaystyle \sum _i\delta W_{M,i}}-{\displaystyle \sum _{k}d{W}_{m,k}}+\\ & {\displaystyle \sum _{k}d{m}_k}\hspace{0.17em}+{\displaystyle \sum _i\delta N_{M,g,i}}+{\displaystyle \sum _k\delta N_{m,g,k}}\end{array}\hspace{1em}\hspace{1em}[\mathrm{U}]$$

(15)

which can be written as two separate differential equations, one for all the merchandise species

$${\displaystyle \sum _{i}d{N}_{M,i}}={\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\delta M_{i,j}}\right)}-{\displaystyle \sum _i\delta W_{M,i}}+{\displaystyle \sum _i\delta N_{M,g,i}}[\mathrm{U}]$$

(16)

and one for all the monetary species k

$${\displaystyle \sum _{k}d{N}_{m,k}}=-{\displaystyle \sum _{k}d{W}_{m,k}}+{\displaystyle \sum _{k}d{m}_k}\hspace{0.17em}+{\displaystyle \sum _k\delta N_{m,g,k}}[\mathrm{U}]$$

(17)

Symbol d is used for the differential of a property (an exact differential), and symbol δ is used for the differential of a non-property (an inexact differential) [1,2,3].

The differential balance Equations (16) and (17) can be themselves split into a set of differential equations, one for each merchandise species i

$$d{N}_{M,i}={\displaystyle \sum _{j=1}^{N_M}\delta M_{i,j}}-\delta W_{M,i}+\delta N_{M,g,i}[\mathrm{U}]$$

(18)

and one for each monetary species k

$$d{N}_{m,k}=-d{W}_{m,k}+d{m}_k\hspace{0.17em}+\delta N_{m,g,k}[\mathrm{U}]$$

(19)

Accounting for merchandise and monetary units (Economic First Law) is intuitive and easily understood. This is not the case with economic entropy (economic Second Law), which, before being accounted for, needs to be defined and understood; it is a process that is much more complex, full of subtleties, and lengthy. Such a development process is also very instructive and useful for looking at Economics through the eyes of Thermodynamics. This results in financial value accounting, which everyone is familiar with.

4. The Economic Second Law, Economic Entropy, and the Economic Entropy Balance Equations

It is well known that some processes occur spontaneously, whereas others only occur if forced. For example, heat flows spontaneously in the direction of decreasing temperature. It is possible to force heat to flow in the increasing temperature direction, but energy needs to be invested for that [1,2,3]. The Energy Conservation Principle embodied by the energy balance equation is thus not enough for a complete description of the thermodynamic systems and processes. Starting from the observations made by the Thermodynamics pioneers on the thermal engines and on the statements made from those observations (Kelvin-Planck and Clausius statements of the Second Law, stating situations impossible to occur), developments can be made leading to the definition of entropy, a new thermodynamic property, and of its generation in irreversible (non-perfect) processes [1,2,3]. Once the property entropy is defined and its generation in irreversible processes is understood, the entropy balance equation is set [1,2,3].

It is also well known that some economic processes occur spontaneously, whereas others only occur if forced. Traded merchandise flows spontaneously in the increasing unit price direction (decreasing economic temperature direction). Traded merchandise can be forced to flow in the direction of decreasing unit price. However, something needs to be invested for such a non-spontaneous economic process to occur. The unit balance equations are thus not enough for a complete description of the economic systems and processes. Starting from the observations made from the economic activity, in a similar way as the Thermodynamics pioneers on the thermal engines, and in particular from the observations of the traded merchandise spontaneous flow in the decreasing economic temperature direction (in the increasing unit price direction), can be stated the economic Kelvin-Planck and Clausius statements of the Second Law. These statements refer to economic processes that cannot occur under normal conditions. Starting from the economic Kelvin-Planck statement of the Second Law, developments are made that allow the definition of merchandise economic entropy and its generation in economically irreversible (non-perfect) processes. Once property merchandise economic entropy is defined and its generation in economically irreversible processes is understood, the economic entropy balance equation can be set.

4.1. Economic Cycle Executed When in Contact with One Merchandise Reservoir

4.1.1. Basic Observations from the Economic Activity

In a similar way as made in Thermodynamics [1,2,3], efforts can be considered to obtain merchandise wealth from an economic engine operating cyclically when in contact with a single merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], as illustrated in Figure 4a. Even if such a cyclic operation is not so common, as the developments leading to the setting of the Second Law of Thermodynamics and entropy are based on cyclic processes, the same approach is also followed in the developments leading to the setting of the economic Second Law and of the economic entropy. $S_{E,M,g}$ [€], the merchandise economic entropy generation, is included in Figure 4 to highlight the economic irreversibility of the analyzed economic cycle, even if the definition and meaning of the merchandise economic entropy and its generation appear only in Section 4.4.2 and Section 4.5.2, respectively.

Some services (labor, electricity, communications, cleaning, security, etc.) are required for any economic activity occurrence. Services are similar to economic friction [6], as they are purchased but lost/dissipated. Lost/dissipated in the sense that once used, they will not be available for any useful purpose. If those units of services need to be paid, as usual, some units of merchandise need to be sold to obtain the monetary units required to pay for those services. This is illustrated in Figure 4b, where the $\left|m\right|$ [U] monetary units need to leave the economic system to pay for the $M_S$ [U] units of services required for the cyclic economic engine operation, which are purchased at the unit price $p_S$ [U/€]. The units of services are not represented in Figure 4b for simplicity. But if the economic system operates cyclically, if $\left|m\right|$ [U] monetary units are leaving the economic system in a cycle it is required to have $m$ [U] monetary units entering the system in a cycle, which are obtained from the selling of some merchandise units $\left|M\right|$ [U] at the unique available unit price $p_1$ [€/U] corresponding to the unique available economic temperature $T_{E,1}$ [U/€], obeying $M_S·p_S=\left|M\right|·p_1=m\times 1$ [€]. To have the economic engine operating cyclically, with the possibility of selling the merchandise units $\left|M\right|$ [U] to the single available merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], those traded merchandise units need to enter the economic system at an infinite economic temperature, as merchandise wealth $W_M=\left|M\right|$ [U] [6]. This possible situation is illustrated in Figure 4b.

If an economic engine operates reversibly, it does not deliver merchandise wealth and does not need to receive merchandise wealth to overcome economic irreversibility. If, instead, the economic engine operates irreversibly, it does not deliver merchandise wealth, and the expected situation in Figure 4a does not occur. On the contrary, it needs to receive merchandise wealth to compensate for internal irreversibility, as illustrated in Figure 4b.

This is analogous to what happens in Thermodynamics [1,2,3], the unique possible situation for a thermal engine that cyclically operates when in contact with a single temperature being to receive mechanical work (which can be understood as heat received at an infinite temperature) to overcome its internal irreversibility, which is dissipated as non-useful heat. The generated non-useful heat is released to the single heat reservoir at the single available temperature. Such a thermal engine thus exhibits the inverse behavior of receiving heat from a single available heat reservoir and releasing mechanical work (the same amount of mechanical work as the heat received from the heat reservoir). If a thermal engine operates reversibly, it does not deliver irreversibility-generated heat and does not require mechanical work to overcome its internal irreversibility [1,2,3].

4.1.2. The Economic Kelvin-Planck and Clausius Statements of the Second Law

The referred economic observations can be expressed in many ways, a possible one being the economic Kelvin-Planck statement of the Second Law [1] ‘No economic system can operate in a cycle and deliver a net amount of merchandise wealth while receiving traded merchandise from a single merchandise reservoir’, what can be mathematically expressed as

$${\displaystyle \underset{T_{E,1}}{\oint }\delta W_M}\le 0[\mathrm{U}]$$

(20)

The equal sign prevails for the reversible economic engine.

Yet not so evident at first sight, another possible statement of the economic observations of the cyclically operating economic engine is the economic Clausius statement of the Second Law [1]: ‘It is impossible for any economic system to operate in such a way that the sole result would be a traded merchandise transfer from a merchandise reservoir at a lower economic temperature to a merchandise reservoir at a higher economic temperature’. The economic Clausius statement is evident from the observations of economic activity, as no natural trading processes occur in such a way that the sole result would be a traded merchandise transfer from a merchandise reservoir at a lower economic temperature to a merchandise reservoir at a higher economic temperature; that is, a traded merchandise transfer from a higher unit price to a lower unit price. This obvious result seems different from the one in Figure 4a, but the equivalence between the economic Kelvin-Planck and Clausius statements of the Second Law can be proved.

4.1.3. Equivalence of the Economic Kelvin-Planck and Clausius Statements of the Second Law

The equivalence of the economic Kelvin-Planck and Clausius statements of the Second Law can be proved with the aid of Figure 5.

Figure 5a represents the situation corresponding to the violation of the economic Clausius statement of the Second Law by system C, in which traded merchandise $\left|M_C\right|$ [U/€] is transferred from the lower economic temperature $T_{E,2}$ [U/€] to the higher economic temperature $T_{E,1}$ [U/€], with no any other effect. On the right-hand side of Figure 5a is an economic engine that operates cyclically when in contact with the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], with $T_{E,1}>T_{E,2}$ [U/€], receiving traded merchandise $M_1$ [U] from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€], releasing the traded merchandise $\left|M_2\right|$ [U] to the merchandise reservoir at the lower economic temperature $T_{E,2}$ [U/€], and releasing the merchandise wealth $W_{M,KP}$ [U]. Cycle KP can be sized such that $\left|M_2\right|=M_C$ [U]. Under these conditions, at the end of one cycle executed by system KP also the composite system $KP+C+T_{E,2}$ executes one cycle when in contact with a single merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], receiving the traded merchandise $\left(M_1-\left|M_C\right|\right)$ [U] from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€] and releasing the merchandise wealth $W_{M,KP}=\left(M_1-\left|M_C\right|\right)$ [U]. Such an economic engine operates cyclically when in contact with a single merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], and releases as merchandise wealth the same traded merchandise it receives, thus violating the economic Kelvin-Planck statement of the Second Law.

Figure 5b represents the situation corresponding to the violation of the economic Kelvin-Planck statement of the Second Law, as the cycle on the right-hand side of Figure 5b releases the merchandise wealth $\left|W_{M,KP}\right|$ [U], when receiving the traded merchandise $M_1=\left|W_{M,KP}\right|$ [U] from the merchandise reservoir at the economic temperature $T_{E,1}$ [U/€]. On the left-hand side of Figure 5b is an economic system that operates cyclically, receiving the merchandise wealth $\left|W_{M,KP}\right|$ [U] released by the KP cycle on the right-hand side of Figure 5b to pump traded merchandise from the merchandise reservoir at the lower economic temperature $T_{E,2}$ [U/€] to the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€]. Invoking the merchandise units balance Equation (18) applied to the $C+KP$ composite economic system in Figure 5b, the net result of the composite system $C+KP$ is the $M_C$ [U] traded merchandise transfer from the merchandise reservoir at the lower economic temperature $T_{E,2}$ [U/€] to the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€], thus violating the economic Clausius statement of the Second Law.

The violation of the economic Clausius statement of the Second Law results in the violation of the economic Kelvin-Planck statement of the Second Law, and the violation of the economic Kelvin-Planck statement of the Second Law results in the violation of the economic Clausius statement of the Second Law, thus proving the equivalence of the economic Kelvin-Planck and Clausius statements of the Second Law.

The economic Clausius statement of the Second Law seems more obvious and evident than the Kelvin-Planck statement of the Second Law. However, developments and results obtained from the economic Kelvin-Planck statement of the Second Law are much more interesting and important.

The structures of Section 4.1.4, Section 4.2.1, Section 4.2.2, Section 4.2.3, Section 4.2.4, Section 4.2.6, Section 4.3, Section 4.4 and Section 4.5 closely follow that in [3] for the well-established Engineering Thermodynamics.

4.1.4. Alternative Form of the Economic Kelvin-Planck Statement of the Second Law

The merchandise units balance Equation (18) applied to the economic system in Figure 4a, with no merchandise units generation, and for just one merchandise species, set that

$$d{N}_M=\delta M-\delta W_M[\mathrm{U}]$$

(21)

whose cyclic integral leads to

$${\displaystyle \oint \delta W_M}={\displaystyle \oint \delta M}[\mathrm{U}]$$

(22)

noting that the cyclic integral of the exact differential $d{N}_M$ [U] is null. The result of Equation (22) considered in Equation (20) leads to

$${\displaystyle \underset{T_{E,1}}{\oint }\delta M}\le 0[\mathrm{U}]$$

(23)

This means that if, due to economic irreversibility, the economic engine needs to receive a given number of merchandise wealth units (a negative value, as it is merchandise wealth entering the economic system), an equal number of traded merchandise units is released (a negative value, as it is traded merchandise leaving the economic system) by the economic engine operating cyclically when in contact with the single merchandise reservoir at the economic temperature $T_{E,1}$ [U/€]. Who, under normal trading conditions, accepts the situation illustrated in Figure 4a, corresponding to receiving traded merchandise $\delta M$ [U] at the economic temperature $T_{E,1}$ [U/€], that is, to purchase traded merchandise paid at the unit price $p_1$ [€/U], and release it as merchandise wealth (and release it at a null unit price)?

Another way to interpret the impossibility of the situation illustrated in Figure 4a is to consider the traded merchandise transfer resulting from an economic temperature difference. If only a single economic temperature is available and not an economic temperature difference (a unit price difference), the driving force for traded merchandise does not exist, the spontaneously traded merchandise transfer does not occur, and it is the spontaneously traded merchandise transfer that allows some of the traded merchandise to be converted into merchandise wealth.

4.1.5. Reversible and Irreversible Economic Cycles

If an economic engine operates cyclically and reversibly when in contact with a single merchandise reservoir, it does not deliver merchandise wealth. However, it does not need to receive merchandise wealth, and the equal sign in Equation (23) prevails. If, instead, the cyclically operating economic engine operates irreversibly (economic internal irreversibility is similar to economic friction, which corresponds to merchandise wealth being dissipated as traded merchandise [6], the economic analog of mechanical work dissipation as heat in Thermodynamics [1,2,3]), it does not deliver merchandise wealth. On the contrary, it needs to receive merchandise wealth to compensate for the economic internal irreversibility, and the unequal sign in Equation (23) prevails.

4.2. Economic Cycle Executed When in Contact with Two Merchandise Reservoirs

The main objective of this section is to extend the result in Equation (23) for a cycle executed when in contact with two merchandise reservoirs. The introduction of Carnot ideas and the Carnot cycle [1,2,3,18] is essential for that.

4.2.1. A First Look at the Economic Carnot Cycle

The economic Carnot cycle, a reversible economic cycle executed by an economic system when in contact with two merchandise reservoirs, is essential for the economic Second Law developments. For the present purposes, it suffices to analyze the economic Carnot cycle operating in contact with the merchandise reservoirs at economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], for $T_{E,1}>T_{E,2}$ [U/€], as schematically illustrated in Figure 6 (what is the same as the economic Carnot cycle operating in contact with the merchandise reservoirs at unit prices $p_1$ [€/U] and $p_2$ [€/U], for $p_1<p_2$ [€/U]). In what follows $M_1$ [U] is the traded merchandise transfer interaction with the $T_{E,1}$ [U/€] merchandise reservoir in one cycle, $M_1={\displaystyle \oint \delta M_1}$ [U], $M_2$ [U] is the traded merchandise transfer interaction with the $T_{E,2}$ [U/€] merchandise reservoir in one cycle, $M_2={\displaystyle \oint \delta M_2}$ [U], and $W_M$ [U] is the merchandise wealth transfer interaction in one cycle, $W_M={\displaystyle \oint \delta W_M}$ [U].

Similarly to the objective of the Carnot thermal engine, of delivering mechanical work when receiving heat from the heat reservoir at a higher temperature, it is assumed that the objective of the economic Carnot engine is to release merchandise wealth, $W_{M,C}>0$ [U], when receiving traded merchandise from the merchandise reservoir at a higher economic temperature $T_{E,1}$ [U/€], being thus $M_{1,C}>0$ [U].

As the economic Carnot cycle is reversible, that means that it can be reversed, and in what concerns the merchandise (traded and wealth) transfer interactions of the direct and reversed cycles, they are related as

$$\left(M_{1,C},M_{2,C},W_{M,C}\right)=-{\left(M_{1,C},M_{2,C},W_{M,C}\right)}_{\mathrm{reversed}}[\mathrm{U}]$$

(24)

4.2.2. On the Merchandise Transfer Interactions of an Unspecified Economic Cycle Executed When in Contact with Two Merchandise Reservoirs

As the economic Carnot engine under analysis delivers merchandise wealth, it is $W_{M,C}>0$ [U], when receiving traded merchandise from the merchandise reservoir at a higher economic temperature $T_{E,1}$ [U/€], that is, for $M_{1,C}>0$. Thus, we are especially interested in investigating what happens with the unspecified economic cycle A in Figure 7a, characterized by the merchandise transfer interactions $\left(M_1,M_2,W_M\right)$ [U], executed when in contact with the same merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€]. This is under the condition that it must release the merchandise wealth $W_M\hspace{0.17em}\left(W_M>0\right)$ [U], when receiving the traded merchandise $M_1\left(M_1>0\right)$ [U] from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€].

The merchandise units balance Equation (18) applied to the unspecified economic cycle A on the left-hand side of Figure 7 gives

$$0=M_1+M_2-W_M[\mathrm{U}]$$

(25)

and from Equation (25), it is $W_M=M_1+M_2$ [U], but it remains to be concluded if it is $M_2>0$ [U] (traded merchandise received by the unspecific economic cycle A) or $M_2<0$ [U] (traded merchandise released by the unspecified economic cycle A).

To search for the possibility of $M_2>0$ [U] consider Figure 7a, where the unspecified economic cycle A, characterized by the merchandise transfer interactions $\left(M_1,M_2,W_M\right)$ [U], operates when in contact with the merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], for $T_{E,1}>T_{E,2}$ [U/€], and the auxiliary economic system B, characterized by the merchandise transfer interactions $\left(M_B,W_{M,B}\right)$ [U], that operates cyclically when in contact with the single merchandise reservoir at the economic temperature $T_{E,2}$ [U/€].

Given that system B executes a cycle when in contact with only the merchandise reservoir at the economic temperature $T_{E,2}$ [U/€], Equation (23) gives

$$M_B\le 0[\mathrm{U}]$$

(26)

As we are searching for the possibility of $M_2>0$ [U], economic cycle B can be sized such that

$$M_B=-M_2[\mathrm{U}]$$

(27)

As the economic systems, A and B execute a cycle when in contact with the merchandise reservoir at the economic temperature $T_{E,2}$ [U/€], noting that the net traded merchandise transfer interaction of the merchandise reservoir at the economic temperature $T_{E,2}$ [U/€] at the end of the cycles executed by the economic systems A and B is $M_2+M_B=0$, as obtained from Equation (27), also the combined system formed by systems, A and B and the merchandise reservoir at the economic temperature $T_{E,2}$ [U/€] executes a cycle when in contact with the single merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], for which Equation (23) imposes $M_1\le 0$ [U]. As all the previous construction was made for the situation when the unspecified cycle A is releasing the merchandise wealth $W_M\hspace{0.17em}\left(W_M>0\right)$ [U] when receiving the traded merchandise $M_1\left(M_1>0\right)$ [U] from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€], assuming that $M_2>0$ [U], if the situation of $M_1\le 0$ [U] is impossible it is also impossible to have $M_2>0$ [U], and it must be $M_2<0$ [U] (traded merchandise released by the unspecified cycle A). The situation of $M_1=0$ [U] is not allowed as it is required that $W_M>0$ [U].

Thus, the merchandise transfer interactions of unspecified economic cycle A executed by the system (the economic engine) in Figure 7a are $\left(\underset{>0}{\underbrace{M_1}},\underset{<0}{\underbrace{M_2}},\underset{>0}{\underbrace{W_M}}\right)$ [U], obeying the following equation that comes from Equation (25)

$$0=\underset{>0}{\underbrace{M_1}}+\underset{<0}{\underbrace{M_2}}-\underset{>0}{\underbrace{W_M}}[\mathrm{U}]$$

(28)

It is thus concluded that the unspecified economic cycle A must be releasing the traded merchandise $\left|M_2\right|\left(M_2<0\right)$ [U] to the merchandise reservoir at the lower economic temperature $T_{E,2}$ [U/€].

4.2.3. Relating the Merchandise Transfer Interactions of the Unspecified Cycle A and the Carnot Cycle Executed When in Contact with the Same Two Merchandise Reservoirs

At this point, it remains to relate merchandise transfer interactions of the unspecified economic cycle A and of the reversible economic Carnot cycle C, both executed when in contact with the same merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], for $T_{E,1}>T_{E,2}$ [U/€], as represented in Figure 7b.

The Carnot cycle C can be sized such that

$$M_1+M_{1,C}=0[\mathrm{U}]$$

(29)

For the situation under analysis $M_1>0$ [U], and it must be $M_{1,C}<0$ [U], that is, considering Equation (25), the economic Carnot cycle in Figure 7b must be the reverse of the direct Carnot cycle corresponding to the unspecified cycle A in Figure 7b. As systems A and C are executing cycles, and considering Equation (25), also the combined system composed of systems A, C, and the merchandise reservoir at the economic temperature $T_{E,1}$ [U/€] executes a cycle when in contact with the single merchandise reservoir at the economic temperature $T_{E,2}$ [U/€], and from Equation (23), it is

$$M_2+M_{2,C}\le 0[\mathrm{U}]$$

(30)

This inequality can be divided by $M_1\left(M_1>0\right)$ [U] or by $-M_{1,C}\left(-M_{1,C}>0\right)$ [U], which have the same absolute value and are both positive given Equation (29) and the text following it, to give

$${\displaystyle \frac{-M_2}{M_1}}\ge {\displaystyle \frac{-M_{2,C}}{M_{1,C}}}[\text{-}]$$

(31)

The result in Equation (31) can be seen as a possible way to express the economic Second Law for a system that operates cyclically when in contact with two merchandise reservoirs. One way to interpret this result is that it sets that the ratio $-M_2/M_1$ [-] is always higher than the limit ratio $-M_{2,C}/M_{1,C}$ [-], this being the ratio corresponding to the (reversible) economic Carnot cycle executed by system C when in contact with the same merchandise reservoirs. In the limit of the equal sign applicability,

$${\displaystyle \frac{-M_2}{M_1}}={\displaystyle \frac{-M_{2,C}}{M_{1,C}}}[\text{-}]$$

(32)

From Equation (28) it is $W_M=M_1+M_2$ [U], setting that $W_M$ [U] depends only on $M_1$ [U] and $M_2$ [U], and from the corresponding merchandise units balance equation for the reversed economic Carnot cycle it is $W_{M,C}=M_{1,C}+M_{2,C}$ [U], setting that $W_{M,C}$ [U] depends only on $M_{1,C}$ [U] and $M_{2,C}$ [U]. Equation (32) applies thus to the situation for which

$$\left(M_{1,C},M_{2,C},W_{M,C}\right)=-\left(M_1,M_2,W_M\right)[\mathrm{U}]$$

(33a)

and considering Equation (24) it is

$$\left(M_1,M_2,W_M\right)=-\left(M_{1,C},M_{2,C},W_{M,C}\right)={\left(M_{1,C},M_{2,C},W_{M,C}\right)}_{\mathrm{reversed}}[\mathrm{U}]$$

(33b)

Thus, in the limiting reversible case, the cycle executed by the unspecified economic system A is the reverse of the economic (reversible) Carnot cycle C and vice versa. The right-hand sides of Equations (31) and (32) can thus be seen as corresponding to the economic reversible cycle executed by the unspecified economic system A. Under these conditions, the reference to the economic Carnot cycle can be avoided, referring only to the reversible cycle executed by the economic system A, writing Equation (31) as

$${\displaystyle \frac{-M_2}{M_1}}\ge {\left({\displaystyle \frac{-M_2}{M_1}}\right)}_{rev}[\text{-}]$$

(34)

The subscript rev refers to the limiting economically reversible cycle executed by economic system A when in contact with the two merchandise reservoirs.

4.2.4. The Absolute Economic Temperature and Merchandise Unit Price Scales

From question to question, the lower limit ${\left(-M_2/M_1\right)}_{rev}$ [-] in the right-hand side of Equation (34) needs now to be related to the involved economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€]. Up to this point, nothing has been specified concerning the sequence of steps forming the economic cycle executed by economic system A, and even economic system A remains unspecified. That means that the unique conditions that have been invoked to arrive at Equation (34) are the involved economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], and it must be

$${\left({\displaystyle \frac{-M_2}{M_1}}\right)}_{rev}=f\left(T_{E,1},T_{E,2}\right)[\text{-}]$$

(35)

where f is an unknown dimensionless function, and $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] are two different numbers, corresponding to the inverse of the unit prices of the involved merchandise reservoirs.

Considering another merchandise reservoir at the economic temperature $T_{E,3}$ [U/€], when the limiting reversible cycle is executed by the unspecified economic system A when in contact with the merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,3}$ [U/€], and the graphical representation in Figure 8, an equation similar to Equation (35) comes

$${\left({\displaystyle \frac{-M_3}{M_1}}\right)}_{rev}=f\left(T_{E,1},T_{E,3}\right)[\text{-}]$$

(36a)

and when the limiting reversible cycle is executed by the unspecified economic system A when in contact with the merchandise reservoirs at the economic temperatures $T_{E,2}$ [U/€] and $T_{E,3}$ [U/€], the equation similar equation to Equation (35) is

$${\left({\displaystyle \frac{-M_3}{-M_2}}\right)}_{rev}=f\left(T_{E,2},T_{E,3}\right)[\text{-}]$$

(36b)

The rule prevailing when writing Equations (35), (36a) and (36b) is that on their left-hand sides are the ratios between the symmetry of the traded merchandise exchanged by the economic system in a cycle with the merchandise reservoir at the lower economic temperature and the traded merchandise exchanged by the economic system in a cycle with the merchandise reservoir at the higher economic temperature, and on their right-hand sides are the functions f of, and in this order, the corresponding economic higher and lower economic temperatures. Figure 8 helps clarification of that rule.

Dividing Equations (36a) and (36b) to eliminate $M_3$ [U], an equation is obtained to express ${\left(-M_2/M_1\right)}_{rev}$ [-], which can be conjugated with Equation (35) to eliminate ${\left(-M_2/M_1\right)}_{rev}$ [-] and obtain that

$$f\left(T_{E,1},T_{E,2}\right)={\displaystyle \frac{f\left(T_{E,1},T_{E,3}\right)}{f\left(T_{E,2},T_{E,3}\right)}}[\text{-}]$$

(37)

As the left-hand side of Equation (37) does not depend on $T_{E,3}$ [U/€], the same must be valid for its right-hand side, and it must be

$$f\left(T_{E,1},T_{E,2}\right)={\displaystyle \frac{\psi \left(T_{E,1}\right)}{\psi \left(T_{E,2}\right)}}[\text{-}]$$

(38)

or making $\varphi \left(T_E\right)=1/\psi \left(T_E\right)$ [U/€] Equation (38) comes

$${\left({\displaystyle \frac{-M_2}{M_1}}\right)}_{rev}={\displaystyle \frac{\varphi \left(T_{E,2}\right)}{\varphi \left(T_{E,1}\right)}}[\text{-}]$$

(39)

Equation (39) can be written to express the ratio between the traded merchandise transfer interactions of the economically reversible cycle that receives the traded merchandise $M_0$ [U] from the merchandise reservoir at the reference economic temperature $T_{E,0}$ [U/€], and releases the merchandise $M$ [U] to the merchandise reservoir at the economic temperature $T_E$ [U/€], as

$$\varphi \left({\theta }_E\right)=\varphi \left({\theta }_{E,0}\right){\left({\displaystyle \frac{-M}{M_0}}\right)}_{rev}[\mathrm{U}/€]$$

(40)

where similarly as in Thermodynamics symbol ${\theta }_E$ [U/€] is being used for the empiric economic temperature, as an absolute economic temperature scale has not yet been set up to this point (symbol $\theta$ [K] is usually used in Thermodynamics to refer to a numerical value of temperature read from a thermometer, the empiric temperature, and not a temperature set from a defined absolute thermodynamic temperature scale [3]). The absolute economic temperature scale definition is based on Equation (40), setting $\varphi \equiv T_E$ [U/€], as

$$T_E\equiv T_{E,0}{\left({\displaystyle \frac{-M}{M_0}}\right)}_{rev}[\mathrm{U}/€]$$

(41)

Such a temperature scale is based on what happens in a reversible cycle, with no conditions or specifications on economic system A and on the sequence of steps forming the economic cycle it executes, which makes it an absolute and universal economic temperature scale. The absolute economic temperature scale depends only on reference point 0, which is the fiducial point of the absolute economic temperature scale. In Thermodynamics, the fiducial point of the absolute temperature scale is the triple point of water, whose absolute temperature assumes the numerical value of 273.16 K [3].

Given Equations (1) and (41), the absolute economic unit price scale, based on the reference point 0, can be defined through the equation

$$p\equiv p_0{\left({\displaystyle \frac{M_0}{-M}}\right)}_{rev}[€/\mathrm{U}]$$

(42)

It is to be noted that previous developments from the beginning of this section could be made using the merchandise unit prices instead of the economic temperatures, setting $\psi \equiv p$ [€/U], leading to Equation (42) defining the absolute economic unit price scale. This is without introducing the auxiliary function $\varphi \left(T_E\right)=1/\psi \left(T_E\right)=T_E=1/p$ [U/€], whose analog is absent in Thermodynamics as there is no physical meaning for the inverse of the absolute temperature, contrary to what happens with the inverse of the economic temperature.

4.2.5. Merchandise Transfer Interactions of the Economic Carnot Cycle

Recalling the main purposes of the developments that follow Equation (34), Equation (41) can be written to relate the traded merchandise transfer interactions of the economic Carnot cycle executed when in contact with the merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], for $T_{E,1}>T_{E,2}$ [U/€], as

$${\left({\displaystyle \frac{-M_2}{M_1}}\right)}_{rev}={\displaystyle \frac{T_{E,2}}{T_{E,1}}}[\text{-}]$$

(43)

or then as

$${\displaystyle \frac{M_{1,C}}{T_{E,1}}}+{\displaystyle \frac{M_{2,C}}{T_{E,2}}}=0[€]$$

(44)

Equation (44) sets the relationship between the traded merchandise interactions of the economic Carnot cycle as depending only on the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€]. From the units balance Equation (18) for the economic Carnot cycle, it is $W_{M,C}=M_{1,C}+M_{2,C}$ [U]. Given that, the merchandise transfer interactions of the economic Carnot cycle depend on the traded merchandise $M_{1,C}$ [U] received from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€] as

$$W_{M,C}=M_{1,C}\left(1-{\displaystyle \frac{T_{E,2}}{T_{E,1}}}\right)=M_{1,C}\left(1-{\displaystyle \frac{p_1}{p_2}}\right)[\mathrm{U}]$$

(45)

$$\left|M_{2,C}\right|=M_{1,C}{\displaystyle \frac{T_{E,2}}{T_{E,1}}}=M_{1,C}{\displaystyle \frac{p_1}{p_2}}[\mathrm{U}]$$

(46)

These results are graphically represented in Figure 9 using the economic Bucher diagram, which also includes a graphical representation of the economic temperature scale through the thicker line.

Given Equation (34), the limiting reversible economic Carnot cycle is represented by the thicker line in Figure 9. Any other allowable situation falls below that limiting line. To the non-reversible cycles executed when in contact with the same merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], for $T_{E,1}>T_{E,2}$ [U/€], for the same $M_1=M_{1,C}$ [U] traded merchandise received from the merchandise reservoir at the higher economic temperature $T_{E,1}$ [U/€], corresponds a merchandise wealth release $W_M<W_{M,C}$ [U], lower than the maximum allowable limit that can be obtained by the economic Carnot cycle, and a traded merchandise release $\left|M_2\right|>\left|M_{2,C}\right|$ [U], whose absolute value is higher than the minimum limit reached by the economic Carnot cycle. The differences $\left(W_M-W_{M,C}\right)<0$ [U] and $\left(\left|M_2\right|-\left|M_{2,C}\right|\right)>0$ [U] are thus a measure of the difference between the best economic performance allowable by a reversible cycle and the economic performance attained by a cycle arbitrary in terms of reversibility, both economic cycles being executed when in contact with the same merchandise reservoirs.

4.2.6. Economic Cycle Executed When in Contact with Two Merchandise Reservoirs

For the present purposes, the conjugation of results in Equations (34) and (43) allows obtaining that

$${\displaystyle \frac{M_1}{T_{E,1}}}+{\displaystyle \frac{M_2}{T_{E,2}}}\le 0[€]$$

(47)

which limiting equality situation corresponds to the economic Carnot cycle executed by the system when in contact with the two merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€].

4.3. Economic Cycle Executed When in Contact with Any Number of Merchandise Reservoirs

Similarly to what is made for the cycle executed by a thermodynamic system when in contact with any number of heat reservoirs, an analog can be made for the economic cycle executed by an economic system when in contact with any number of merchandise reservoirs.

Looking at Equation (47), and particularly to its form, as the absolute economic temperature $T_{E,1}$ [U/€] is a positive definite value, the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing an economic cycle when in contact with the single economic temperature $T_{E,1}$ [U/€], Equation (23), can also be written as

$${\displaystyle \frac{M_1}{T_{E,1}}}\le 0[€]$$

(48)

Equation (47) sets the result of the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing an economic cycle when in contact with two merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€], and from these two equations, it seems that the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing a cycle when in contact with $n$ merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€], $T_{E,2}$ [U/€], …, $T_{E,n}$ [U/€] must result in

$${\displaystyle \sum _{j=1}^n\frac{M_j}{T_{E,j}}}\le 0[€]$$

(49)

However, this result must be confirmed. The mathematical induction method will be used for that, as made by Bejan [3] for the well-established Thermodynamics.

Equation (48) sets the result of the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing an economic cycle when in contact with one merchandise reservoir at the economic temperature $T_{E,1}$ [U/€]. It is assumed that the result of the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing an economic cycle when in contact with the $n$ merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€], $T_{E,2}$ [U/€], …, $T_{E,n}$ [U/€] is that in Equation (49). If from the result in Equation (49), it can be proved that the result of the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing a cycle when in contact with the $n+1$ merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€], $T_{E,2}$ [U/€], …, $T_{E,n+1}$ [U/€] it is

$${\displaystyle \sum _{j=1}^{n+1}\frac{M_j}{T_{E,j}}}\le 0[€]$$

(50)

then, the result expressed by Equation (49) is proven. The proof of the result in Equation (50) from the result in Equation (49) is obtained with the aid of Figure 10. Note that in Figure 10 the merchandise reservoirs are at the middle height of the figure, and the economic systems executing an economic cycle when in contact with these merchandise reservoirs are the larger rectangles at the bottom and top of the figure.

Economic system $\left(A\right)$ executes an economic cycle when in contact with the $\left(T_{E,1}\right)$ [U/€], $\left(T_{E,2}\right)$ [U/€], …, $\left(T_{E,n}\right)$ [U/€] and $\left(T_{E,n+1}\right)$ [U/€] merchandise reservoirs. In turn, the composite system formed by the economic system $\left(A\right)$, the merchandise reservoir $\left(T_{E,n+1}\right)$ [U/€], and the economic system $\left(C\right)$ execute a cycle when in contact with the $\left(T_{E,1}\right)$ [U/€], $\left(T_{E,2}\right)$ [U/€], …, $\left(T_{E,n}\right)$ [U/€] merchandise reservoirs. For the present purposes, the economic Carnot cycle $\left(C\right)$ can be sized such that

$$M_{\left(n+1\right)}+M_{\left(n+1\right),C}=0[\mathrm{U}]$$

(51)

a condition that does not impose any restricting condition over what is to be proved but that allows the involved cycles to be related, as expected.

Economic system $\left(C\right)$ is a reversible economic (Carnot) cycle, for which it is

$${\displaystyle \frac{M_{\left(n\right),C}}{T_{E,n}}}+{\displaystyle \frac{M_{\left(n+1\right),C}}{T_{E,n+1}}}=0[€]$$

(52)

No restrictions are imposed on the direction of traded merchandise transfer interactions.

The Second Law equation for the $\left(A\right)$ + $\left(T_{n+1}\right)$ + $\left(C\right)$ composite economic system in Figure 10, which executes a cycle when in contact with the $\left(T_{E,1}\right)$ [U/€], $\left(T_{E,2}\right)$ [U/€], …, $\left(T_{E,n}\right)$ [U/€] merchandise reservoirs, may be written as

$${\displaystyle \sum _{j=1}^n\frac{M_j}{T_{E,j}}}+{\displaystyle \frac{M_{\left(n\right),C}}{T_{E,n}}}\le 0[€]$$

(53)

and using Equations (52) and (53) can be rewritten as

$${\displaystyle \sum _{j=1}^n\frac{M_j}{T_{E,j}}}-{\displaystyle \frac{M_{\left(n+1\right),C}}{T_{E,n+1}}}\le 0[€]$$

(54)

Equation (51) can be used to obtain $M_{\left(n+1\right),C}=-M_{\left(n+1\right)}$ [U], a result that can be used to write Equation (54) as

$${\displaystyle \sum _{j=1}^n\frac{M_j}{T_{E,j}}}+{\displaystyle \frac{M_{\left(n+1\right)}}{T_{E,n+1}}}={\displaystyle \sum _{j=1}^{n+1}\frac{M_j}{T_{E,j}}}\le 0[€]$$

(55)

This proves the result in Equation (49), which is the economic Kelvin-Planck statement of the Second Law for an unspecified economic system executing an economic cycle when in contact with $n$ merchandise reservoirs at the economic temperatures $T_{E,1}$ [U/€], $T_{E,2}$ [U/€], …, $T_{E,n}$ [U/€]. Equation (49) can thus be seen as the generalization for $n$ of the result in Equation (48) for $n=1$.

4.4. Economic Cycle Executed When in Contact with an Infinite Number of Merchandise Reservoirs

4.4.1. The Economic Clausius Inequality

Similarly to what is made for the cycle executed by a thermodynamic system when in contact with an infinite number of heat reservoirs [3], it can be obtained for the economic cycle executed by the economic system when in contact with an infinite number of merchandise reservoirs that

$${\displaystyle \oint \frac{\delta M}{T_E}}\le 0[€]$$

(56)

Infinitesimal traded merchandise $\delta M$ [U] is transferred where the system’s boundary economic temperature is $T_E$ [U/€]. Equation (56) can be referred to as the economic Clausius inequality. It sets that for the economically irreversible cycles, the absolute value of the negative contributions of $\delta M/T_E$ [€] to the cyclic integral ${\displaystyle \oint \left(\delta M/T_E\right)}$ [€] surpass the positive contributions of $\delta M/T_E$ [€] to this same cyclic integral. Under the reversible conditions, for which prevails the equal sign of Equation (56), the economic Clausius inequality sets that the negative contributions of $\delta M/T_E$ [€] to the cyclic integral ${\displaystyle \oint \left(\delta M/T_E\right)}$ [€] cancel the positive contributions of $\delta M/T_E$ [€] to this same cyclic integral.

4.4.2. Defining Economic Entropy

Equation (56) is of major relevance as for the limiting economic reversible cycle, the equal sign prevails, and it is

$${{\displaystyle \oint \left(\frac{\delta M}{T_E}\right)}}_{rev}=0[€]$$

(57)

If the cyclic integral is null, that means the integrand is an exact differential, a property of the economic system, thus allowing the definition of the property merchandise economic entropy through its differential form as

$$d{S}_{E,M}\equiv {\left({\displaystyle \frac{\delta M}{T_E}}\right)}_{rev}[€]$$

(58)

where subscript rev indicates that this definition is valid only for an economic reversible cycle. Section 4.8.6 includes additional considerations on Equation (58), defining the merchandise economic entropy.

Given the definition of the economic temperature in Equations (1) and (6), it is now clear, as obtained by demonstration, based on the traded merchandise and the economic temperature, that $d{S}_{E,M}\equiv {\left(\delta M/T_E\right)}_{rev}={\left(\delta M·p\right)}_{rev}$ [€], that the merchandise economic entropy $S_{E,M}$ [€] of an economic system is the financial value of the merchandise in the economic system.

As the merchandise economic entropy $S_{E,M}$ [€] is a property of the economic system, its change on a 1-2 economic process depends only on the starting and ending states 1 and 2, and not on the nature of the 1-2 economic process, and it is

$${\displaystyle \underset{1}{\overset{2}{\int }}d{S}_{E,M}}=S_{E,M,2}-S_{E,M,1}[€]$$

(59)

Rudolf Clausius coined the word entropy to be similar to energy but also containing the Greek word ‘τρόπος’ (tropos), which means ‘a way’, entropy being the abbreviation of the conjugation of ‘energy’ and ‘tropos’ [3]. In the economic context, something parallel happens, as the merchandise economic entropy is similar to the merchandise units associated with the ‘one way’ traded merchandise transfers taking place in trading operations, traded merchandise transfers occurring in the decreasing economic temperature direction.

4.5. Economic Process When in Contact with Any Number of Merchandise Reservoirs

The cyclic process is abandoned in this section, obtaining results that apply to a process that can or cannot be a cycle.

4.5.1. An Economic Process as Part of an Economic Cycle

Similarly to what is made for the thermodynamic process when in contact with any number of heat reservoirs [3], it can be obtained from Equation (56) for the economic cycle 1-2-1 executed by the economic system when in contact with an infinite number of merchandise reservoirs, as illustrated in Figure 11, that

$${\displaystyle \underset{1}{\overset{2}{\int }}\frac{\delta M}{T_E}}+{\displaystyle \underset{2}{\overset{1}{\int }}\frac{\delta M}{T_E}}\le 0[€]$$

(60)

4.5.2. Defining the Merchandise Economic Entropy Generation

As illustrated in Figure 11, the part 2-1 of the economic cycle corresponds to an economic reversible process, and Equation (60) can be written as

$$\left(S_{E,M,2}-S_{E,M,1}\right)-{\displaystyle \underset{1}{\overset{2}{\int }}\frac{\delta M}{T_E}}\ge 0[€]$$

(61)

the magnitude of the inequality in Equation (61) defining the merchandise economic entropy generation in the 1-2 process as

$$S_{E,M,g}\equiv \left(S_{E,M,2}-S_{E,M,1}\right)-{\displaystyle \underset{1}{\overset{2}{\int }}\frac{\delta M}{T_E}}\ge 0[€]$$

(62)

The merchandise economic entropy generation is not a property of the economic system, as it depends on the 1-2 economic process, and it is a measure of the 1-2 economic process irreversibility (departure from economic equilibrium) or the 1-2 economic process imperfection.

The result in Equation (62) can be seen as a way of writing the merchandise economic entropy balance equation for an economic system evolving between states 1 and 2. If written on a time rate basis, for the instantaneous accounting of the time rate of merchandise economic entropy generation, and in a slightly different form similar to the units balance equations in Section 3, it can be written as

$${\displaystyle \frac{d{S}_{E,M}}{dt}}={\displaystyle \sum _{j=1}^{N_M}\frac{{\dot{M}}_j}{T_{E,j}}}+{\dot{S}}_{E,M,g}[€/\mathrm{s}]$$

(63)

where ${\dot{S}}_{E,M,g}\ge 0$ [€/s]. In Equation (63) it is assumed that the economic system is in economic thermal contact with the $N_M$ merchandise reservoirs, instantaneously exchanging the traded merchandise flow rate ${\dot{M}}_j$ [U/s] with the j merchandise reservoir at the economic temperature $T_{E,j}$ [U/€], the traded merchandise transfer interaction flow rate ${\dot{M}}_j$ [U/s] entering (${\dot{M}}_j>0$ [U/s]) or leaving (${\dot{M}}_j<0$ [U/s]) the economic system through the portion of its boundary where the economic temperature is $T_{E,j}$ [U/€].

Merchandise economic entropy is a property of the economic system, and monetary economic entropy is another property. The (merchandise and monetary) economic entropy generation is a measure of the irreversibility (imperfection) of an economic process. This in a way parallel to that followed in Thermodynamics to define entropy as a property of a thermodynamic system and the entropy generation as a measure of the irreversibility (imperfection) of a thermodynamic process [3].

The next step is to set the economic entropy (Second Law) balance equations. Section 4.6 and Section 4.7 follow closely Sections 3.1 and 3.2 of [6], respectively.

4.6. Meaning of the New Concepts and Variables Involved in the Economic Entropy Balance Equations

4.6.1. A New Understanding of the Economic Entropy as the Financial Value

The economic entropy $S_E$ [€] of an economic system is its financial value. If an economic system is composed of $N_{M,i}$ [U] merchandise units of species i, each with its unit price $p_i$ [€/U], its merchandise economic entropy is

$$S_{E,M}={\displaystyle \sum _i{N}_{M,i}·p_i}[€]$$

(64)

If the same economic system is also composed of $N_{m,k}$ [U] monetary units of species k, each with the exchange rate $F_k$ [€/U], its monetary economic entropy is

$$S_{E,m}={\displaystyle \sum _k{N}_{m,k}·F_k}[€]$$

(65)

The economic entropy of the system is thus

$$S_E=S_{E,M}+S_{E,m}={\displaystyle \sum _i{N}_{M,i}·p_i}+{\displaystyle \sum _k{N}_{m,k}·F_k}[€]$$

(66)

These equations have been previously introduced in Section 3.1, referring to financial value and not economic entropy, as economic entropy has only been defined and understood as the financial value of an economic system in Section 4.4.

4.6.2. Merchandise and Monetary Economic Entropy Accumulation Rates in the Economic System

The economic entropy accumulation rate in the economic system is the sum of the merchandise and monetary economic entropy accumulation rates in the system

$${\displaystyle \frac{d{S}_E}{dt}}={\displaystyle \frac{d{S}_{E,M}}{dt}}+{\displaystyle \frac{d{S}_{E,m}}{dt}}[€/\mathrm{s}]$$

(67)

4.6.3. Merchandise Economic Entropy Flow Rates

As mentioned in Section 2.1, the relevance of the economic temperature of a given merchandise species is mainly on how it influences the traded merchandise economic entropy flow rates entering or leaving the economic system through its boundary. This is why each traded merchandise flow rate ${\dot{M}}_i$ [U/s] of species i is associated with the economic temperature $T_{E,i}$ [U/€] at which it crosses the economic system’s boundary, as illustrated in Figure 2 and Figure 12.

The merchandise economic entropy flow rate associated with the traded merchandise transfer flow rate $\dot{M}$ [U/s] is

$${\displaystyle \frac{\dot{M}}{T_E}}=\dot{M}·p[€/\mathrm{s}]$$

(68)

which is the traded merchandise financial value flow rate entering ($\dot{M}>0$ [U/s]) or leaving ($\dot{M}<0$ [U/s]) the economic system through its boundary, where the economic temperature is $T_E$ [U/€].

Monetary values are fixed amounts of monetary economic entropy (fixed financial values), which do not change in trading operations, giving them a different nature when compared with traded merchandise economic entropy flow rates.

4.6.4. Monetary Economic Entropy Flow Rates

The financial value flow rate of the exchanged monetary flow rate ${\dot{m}}_k$ [U/s] is its economic entropy flow rate

$${\displaystyle \frac{{\dot{m}}_k}{T_{E,m,k}}}={\displaystyle \frac{{\dot{m}}_k}{1/F_k}}={\dot{m}}_k·F_k[€/\mathrm{s}]$$

(69)

which has thus a nature different from that of the merchandise economic entropy flow rate of the traded merchandise flow rate $\dot{M}$ [U/s]. Even thus, for economic entropy balances, the exchanged monetary flow rates ${\dot{m}}_k$ [U/s] can be seen as exchanged at the fixed/constant economic temperature $T_{E,m,k}=1/F_k$ [U/€], as illustrated in Figure 2 and Figure 12. If the $m_k$ [U] monetary units are already expressed in Euros, $F_k=1$ [€/U] and $T_{E,m,k}=1/1=1$ [U/€].

Once the meanings of the introduced concepts and variables for looking at Economics through the eyes of Thermodynamics are clarified, the economic entropy balance equations can be introduced and discussed.

4.7. The Economic Entropy Balance Equations

4.7.1. General Form of the Economic Entropy Balance Equation

Written in the same form as usual in Engineering Thermodynamics [1,2,3], the economic entropy balance equation sets that

$$\begin{array}{ll}\underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{\left(\begin{array}{l}\mathrm{time}\mathrm{rate}\mathrm{of}\mathrm{change}\\ \mathrm{of}\mathrm{the}\mathrm{merchandise}\\ \mathrm{and}\mathrm{monetary}\mathrm{economic}\\ \mathrm{entropy}\mathrm{in}\mathrm{the}\mathrm{system}\end{array}\right)}}=& \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{net}\mathrm{flow}\mathrm{rate}\mathrm{of}\mathrm{traded}\\ \mathrm{merchandise}\mathrm{economic}\\ \mathrm{entropy}\mathrm{entering}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}+\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{flow}\mathrm{rate}\mathrm{of}\mathrm{exchanged}\\ \mathrm{monetary}\mathrm{economic}\\ \mathrm{entropy}\mathrm{entering}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}-\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{\left(\begin{array}{l}\mathrm{flow}\mathrm{rate}\mathrm{of}\mathrm{exchanged}\\ \mathrm{monetary}\mathrm{economic}\\ \mathrm{entropy}\mathrm{leaving}\mathrm{the}\\ \mathrm{system}\end{array}\right)}}+\\ & \underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{\left(\begin{array}{l}\mathrm{net}\mathrm{merchandise}\\ \mathrm{economic}\mathrm{entropy}\\ \mathrm{generation}\mathrm{rate}\\ \mathrm{in}\mathrm{the}\mathrm{system}\end{array}\right)+\left(\begin{array}{l}\mathrm{net}\mathrm{monetary}\\ \mathrm{economic}\mathrm{entropy}\\ \mathrm{generation}\mathrm{rate}\\ \mathrm{in}\mathrm{the}\mathrm{system}\end{array}\right)}}\end{array}\hspace{1em}\hspace{1em}[€/\mathrm{s}]$$

It is well-known in Thermodynamics that mechanical work rates have associated null entropy flow rates [1,2,3]. In what follows, it is explained why, similarly, the wealth (merchandise and monetary) flow rates entering or leaving an economic system through its boundary have associated null economic entropy flow rates, the reason why the (merchandise and monetary) wealth contributions are absent from the economic entropy balance equations.

4.7.2. Economic Entropy Balance Equation for All the Merchandise and Monetary Species

As illustrated in Figure 12, detailing, the economic entropy balance equation sets that the time rate of change of the merchandise and monetary economic entropy of all the merchandise species i and of all the monetary species k in the economic system equals the sum of the traded merchandise economic entropy flow rates ${\dot{M}}_{i,j}/T_{E,i,j}={\dot{M}}_{i,j}·p_{i,j}$ [€/s] of all the merchandise species i exchanged by the economic system with the $N_M$ merchandise reservoirs, plus the sum of the monetary economic entropy flow rates ${\dot{m}}_k·F_k$ [€/s] of all the monetary species k entering the economic system, minus the sum of the monetary economic entropy flow rates ${\dot{m}}_k·F_k$ [€/s] of all the monetary species k leaving the economic system, plus the sum of the merchandise and monetary economic entropy generation rates ${\dot{S}}_{E,M,g,i}$ [€/s] and ${\dot{S}}_{E,m,g,k}$ [€/s] of all the merchandise and monetary species in the economic system, that is

$$\begin{array}{ll}\underset{\mathrm{in}\mathrm{the}\mathrm{system}}{\underbrace{{\displaystyle \sum _i\left(\frac{d{S}_{E,M,i}}{dt}\right)}+{\displaystyle \sum _k\left(\frac{d{S}_{E,m,k}}{dt}\right)}}}=& \underset{\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{through}\mathrm{the}\\ \mathrm{system}’\mathrm{s}\mathrm{boundary}\end{array}}{\underbrace{{\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\frac{{\dot{M}}_{i,j}}{T_{E,i,j}}}\right)}}}+\\ & \underset{\mathrm{through}\mathrm{the}\mathrm{system}’\mathrm{s}\mathrm{boundary}}{\underbrace{{\displaystyle \sum _k\left({\displaystyle \sum _{in}{\dot{m}}_k·F_k}\right)}-{\displaystyle \sum _k\left({\displaystyle \sum _{out}{\dot{m}}_k·F_k}\right)}}}\\ & \underset{\mathrm{in}\hspace{1em}\mathrm{the}\mathrm{system}}{\underbrace{{\displaystyle \sum _i{\dot{S}}_{E,M,g,i}}+{\displaystyle \sum _k{\dot{S}}_{E,m,g,k}}}}\end{array}[€/\mathrm{s}]$$

(70)

4.7.3. Separated Economic Entropy Balance Equations for the Merchandise and Monetary Species

The entropy balance Equation (70) can be written as two separate equations, one for all the merchandise species

$${\displaystyle \sum _i\left(\frac{d{S}_{E,M,i}}{dt}\right)}={\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\frac{{\dot{M}}_{i,j}}{T_{E,i,j}}}\right)}+{\displaystyle \sum _i{\dot{S}}_{E,M,g,i}}[€/\mathrm{s}]$$

(71)

and one for all the monetary species

$${\displaystyle \sum _k\left(\frac{d{S}_{E,m,k}}{dt}\right)}={\displaystyle \sum _k\left({\displaystyle \sum _{in}{\dot{m}}_k·F_k}\right)}-{\displaystyle \sum _k\left({\displaystyle \sum _{out}{\dot{m}}_k·F_k}\right)}+{\displaystyle \sum _k{\dot{S}}_{E,m,g,k}}[€/\mathrm{s}]$$

(72)

As mentioned in Section 3.3, the separation of Equation (70) in Equations (71) and (72) can be made as in the economic operations there are no conversions of merchandise units into monetary units, nor are there conversions of monetary units into merchandise units.

4.7.4. Relating the Merchandise and Economic Entropy Balance Equations in Trading Operations

Merchandise units are exchanged by monetary units in trading operations, but one unit of one is not converted into units of the other. Even if Equations (71) and (72) can be considered separately, they are coupled in trading operations, as referred to in Section 3.3.4. Purchasing (inlet) of $M_p$ [U] merchandise units at the unit price $p_p$ [€/U] corresponds to the inlet traded merchandise economic entropy $M_p/T_{E,p}=M_p·p_p$ [€], and to the associated outlet monetary economic entropy $\left|m_p\right|\times 1$ [€] that corresponds to the monetary financial value spent in the purchasing operation, obeying $M_p·p_p=\left|m_p\right|\times 1$ [€]. Selling (outlet) of $\left|M_s\right|$ [U] merchandise units at the unit price $p_s$ [€/U] corresponds to the outlet traded merchandise economic entropy $\left|M_s\right|/T_{E,s}=\left|M_s\right|·p_s$ [€], and to the associated inlet monetary economic entropy $m_s\times 1$ [€] that corresponds to the monetary financial value received from the traded merchandise selling operation, obeying $\left|M_s\right|·p_s=m_s\times 1$ [€]. It is through relations of the type

$$\begin{array}{l}\{\begin{array}{l}{M}_p·p_p=\left|m_p\right|\times 1\\ \left|M_s\right|·p_s=m_s\times 1\end{array}\end{array}[€]$$

(73)

that Equations (71) and (72) are coupled. This same result was previously anticipated in Equation (12), based only on financial arguments, before introducing, defining, and accounting for the merchandise and monetary economic entropy. Multiplications by 1 are retained to highlight that the monetary units $m$ [U] in Euros are associated with the monetary economic entropy $m/T_{E,m}=m/1$ [€], which may be written as $m\left(1/T_{E,m}\right)=m\times 1$ [€].

4.7.5. Economic Entropy Balance Equations for a Single Merchandise Species and a Single Monetary Species

Looking at Equation (71) it can be decomposed into a set of equations, one for each merchandise species i

$${\displaystyle \frac{d{S}_{E,M,i}}{dt}}={\displaystyle \sum _{j=1}^{N_M}\frac{{\dot{M}}_{i,j}}{T_{E,i,j}}}+{\dot{S}}_{E,M,g,i}[€/\mathrm{s}[$$

(74)

and Equation (72) can be decomposed into a set of equations, one for each monetary species k

$${\displaystyle \frac{d{S}_{E,m,k}}{dt}}={\displaystyle \sum _{in}{\dot{m}}_k·F_k}-{\displaystyle \sum _{out}{\dot{m}}_k·F_k}+{\dot{S}}_{E,m,g,k}[€/\mathrm{s}]$$

(75)

This can be made because there are no crossed relations between them, even if in trading operations they are coupled through equations of the type of Equation (73).

4.7.6. Notes on the Traded Merchandise and Merchandise Wealth Flow Rates

The traded merchandise flow rate $\dot{M}$ [U/s] crossing the system’s boundary at the economic temperature $T_E=1/p$ [U/€] has the merchandise economic entropy flow rate $\dot{M}/T_E=\dot{M}·p$ [€/s]. The merchandise wealth flow rate ${\dot{W}}_M$ [U/s] crossing the economic system’s boundary has a null economic entropy flow rate. This is like the merchandise wealth flow rate ${\dot{W}}_M$ [U/s] is a traded merchandise flow rate crossing the economic system’s boundary at an infinite economic temperature. From the merchandise units viewpoint, it continues to be the merchandise wealth flow rate ${\dot{W}}_M$ [U/s], and from the merchandise economic entropy viewpoint it has the null merchandise economic entropy flow rate ${\left({\dot{W}}_M/T_E\right)}_{T_E\to \infty }=0$ [€/s] or, given Equation (1), the merchandise wealth crosses the system’s boundary at a null unit price, ${\left({\dot{W}}_M·p\right)}_{p\to 0}=0$ [€/s].

This is similar to what happens in Thermodynamics [1,2,3], as a mechanical work rate $\dot{W}$ [J/s] can be seen as a heat flow rate $\dot{Q}$ [J/s] crossing the thermodynamic system’s boundary at an infinite temperature. From the energy transfer viewpoint, it continues to be the work rate $\dot{W}$ [J/s], and from the entropy viewpoint it has the null entropy flow rate ${\left(\dot{W}/T\right)}_{T\to \infty }=0$ [J/(s·K)].

4.7.7. Notes on the Traded Merchandise and Merchandise Wealth

The energy from the sun or wind is available at null unit prices. These are units of energy that are thus available as merchandise wealth. However, once harvested solar and wind energy is placed in the market, giving it a non-zero unit price, merchandise wealth is converted into traded merchandise, and (in principle) traded merchandise will no longer be converted back into merchandise wealth (units of energy available at null unit prices). Imagination, ideas, time, and previous and acquired knowledge are available for free at null unit prices, and are thus available as merchandise wealth.

It is usual to say that the (true) wealth corresponds to things that are not in the market, that have no price, and/or that are not to be sold. In the present context, they have a null unit price, which is the same as saying that they have an infinite economic temperature.

4.7.8. Evaluation of the Economic Entropy Generation

The economic entropy balance Equations (70)–(72), (74) and (75) can be used for many purposes. They may be used especially for the evaluation of the economic entropy generation rates (the last terms on their right-hand sides) or, in a much more common language, for the evaluation of the financial value generation rates in the economic processes if all the remaining terms in those equations are already known.

Compared with the Second Law of Thermodynamics [1,2,3], for which irreversibility and entropy generation are associated with imperfection in the operation of the thermodynamic systems, the developed and proposed economic Second Law states that economic irreversibility or economic imperfection is associated with economic entropy generation (financial value generation).

4.7.9. Differential Form of the Economic Entropy Balance Equations

Multiplying Equation (70) by dt allows obtaining the differential form of the economic entropy balance equation for an infinitesimal process as

$$\begin{array}{ll}{\displaystyle \sum _{i}d{S}_{E,M,i}}+{\displaystyle \sum _{k}d{S}_{E,m,k}}=& {\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\frac{\delta M_{i,j}}{T_{E,i,j}}}\right)}+\\ & {\displaystyle \sum _{k}d{m}_k·F_k}+{\displaystyle \sum _i\delta S_{E,M,g,i}}+{\displaystyle \sum _k\delta S_{E,m,g,k}}\end{array}[€]$$

(76)

which can be split into two differential equations, one for all the merchandise species i

$${\displaystyle \sum _{i}d{S}_{E,M,i}}={\displaystyle \sum _i\left({\displaystyle \sum _{j=1}^{N_M}\frac{\delta M_{i,j}}{T_{E,i,j}}}\right)}+\delta S_{E,M,g,i}[€]$$

(77)

and one for all the monetary species k

$${\displaystyle \sum _{k}d{S}_{E,m,k}}={\displaystyle \sum _{k}d{m}_k·F_k}+{\displaystyle \sum _k\delta S_{E,m,g,k}}[€]$$

(78)

Equation (77) may be split into a set of differential equations, one for each merchandise species i

$$d{S}_{E,M,i}={\displaystyle \sum _{j=1}^{N_M}\frac{\delta M_{i,j}}{T_{E,i,j}}}+\delta S_{E,M,g,i}[€]$$

(79)

and Equation (78) can be split into a set of differential equations, one for each monetary species k

$$d{S}_{E,m,k}=d{m}_k·F_k+\delta S_{E,m,g,k}[€]$$

(80)

4.8. Traded Merchandise Transfer Through a Finite Economic Temperature Difference

The (merchandise and monetary units and economic entropy) balance equations are considered on a time rate basis for the present purposes. Section 4.8.1 and Section 4.8.4 closely follow Section 5.1 of [6].

4.8.1. Traded Merchandise Transfer through a Finite Economic Temperature Difference

The process under analysis consists of the steady merchandise transfer flow rate $\left|\dot{M}\right|$ [U/s] through the economic temperature difference $\left(T_{E,1}-T_{E,2}\right)$ [U/€], $T_{E,1}>T_{E,2}$ [U/€], without obtaining any merchandise wealth rate, as schematically represented in Figure 13.

The merchandise units balance Equation (13) and the merchandise economic entropy balance Equation (74) are applied to the economic system sandwiched between the $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] merchandise reservoirs in Figure 13 to give, respectively, $0=\dot{M}-\left|\dot{M}\right|$ [U/s] and $0=\dot{M}/T_{E,1}-\left|\dot{M}\right|/T_{E,2}+{\dot{S}}_{E,M,g}$ [€/s]. The merchandise economic entropy generation rate due to the irreversible merchandise transfer across the $\left(T_{E,1}-T_{E,2}\right)$ [U/€] economic temperature difference is thus

$${\dot{S}}_{E,M,g}=\dot{M}\left({\displaystyle \frac{1}{T_{E,2}}}-{\displaystyle \frac{1}{T_{E,1}}}\right)=\dot{M}\left(p_2-p_1\right)[€/\mathrm{s}]$$

(81)

4.8.2. A Better Understanding of the Merchandise Economic Entropy and Its Generation

The merchandise flow rate $\dot{M}$ [U/s] is traded, the purchasing unit price being $p_1=1/T_{E,1}$ [€/U] and the selling unit price being $p_2=1/T_{E,2}$ [€/U], $p_2>p_1$ [€/U]. The rate of profit generation (the rate of financial value generation) in the trading operation, which is the same as the rate of merchandise economic entropy generation, as given by Equation (81), is a different way to look at a well-understood and familiar result: the financial profit rate obtained from the trading process is the product of the traded merchandise flow rate and the difference between the selling and purchasing unit prices.

Equation (81) is the main result leading to the definition of the economic temperature as the inverse of the unit price, as anticipated in Equation (1). Considering a Society familiar with the numbers of units, with units [U], and with the economic entropy with units [€], but not with the economic temperature $T_E$ [U/€], from Equation (81), it can be learned that the units of the economic temperature are 1/[€/U] = [U/€]. In Thermodynamics, Society is familiar with energy, with units [J], and with absolute temperature, with units [K], the definition expression of entropy as $dS\equiv {\left(\delta Q/T\right)}_{rev}$ [J/K] teaching that entropy has units [J/K] [1,2,3].

4.8.3. Monetary Transfers Associated with the Traded Merchandise Transfer through a Finite Economic Temperature Difference

The monetary units balance Equation (14) and the monetary economic entropy balance Equation (75) applied to the same system in Figure 13 give, respectively,

$${\displaystyle \frac{d{N}_m}{dt}}=-\left|{\dot{m}}_1\right|+{\dot{m}}_2[\mathrm{U}/\mathrm{s}]$$

(82)

$${\displaystyle \frac{d\left(N_m\times 1\right)}{dt}}=-\left|{\dot{m}}_1\right|\times 1+{\dot{m}}_2\times 1=\left({\dot{m}}_2-\left|{\dot{m}}_1\right|\right)\times 1[€/\mathrm{s}]$$

(83)

Equations (82) and (83) are coupled with the traded merchandise flow rate $\left|M\right|$ [U], through the conditions

$$\{\begin{array}{l}\dot{M}·p_1=\left|{\dot{m}}_1\right|\times 1\\ \left|\dot{M}\right|·p_2={\dot{m}}_2\times 1\end{array}[€/\mathrm{s}]$$

(84)

The economic system sandwiched between the $T_{E,1}$ [U/€] and $T_{E,2}$ [U/€] merchandise reservoirs in Figure 13 operates steadily from the merchandise point of view, thus being $d{N}_M/dt=0$ [U/s] and $d{S}_{E,M}/dt=0$ [€/s]. By its turn, the monetary flow rate ${\dot{m}}_2$ [U/s] entering the economic system is higher than the monetary flow rate $\left|{\dot{m}}_1\right|$ [U/s] leaving it, thus leading to a positive monetary accumulation rate in the economic system, $d{N}_m/dt=\left({\dot{m}}_2-\left|{\dot{m}}_1\right|\right)>0$ [U/s]. This, in turn, leads to a positive accumulation rate of monetary economic entropy in the system, $\left(d{N}_m/dt\right)\times 1=\left({\dot{m}}_2-\left|{\dot{m}}_1\right|\right)\times 1>0$ [€/s]), which is the same as saying that it leads to a positive accumulation rate of monetary financial value in the economic system. This is the monetary profit generation rate of the analyzed trading operation.

4.8.4. Why Can Not Traded Merchandise Be Integrally Converted into Merchandise Wealth in a Cycle?

Once the economic entropy and the merchandise economic entropy generation are defined, and the economic entropy balance equations are set, it can be easily analyzed why traded merchandise cannot be integrally converted into merchandise wealth in a cycle.

The analysis begins with the cycle executed by the economic system when in contact with a single merchandise reservoir. The question under analysis arises in Section 4.1.1. If the $M_1$ [U] traded merchandise units entering the economic system at the finite economic temperature $T_{E,1}$ [U/€] would eventually be integrally converted into the same number of wealth merchandise units, as illustrated in Figure 4a, the merchandise economic entropy balance Equation (74) for the economic system in Figure 4a would give $S_{E,M,g}=-M_1/T_{E,1}=-M_1·p_1\le 0$ [€], a negative merchandise economic entropy generation. As merchandise economic entropy generation can only be positive (no natural merchandise trading operations exist to generate negative financial value), this indicates that it is impossible, even for the most perfect (reversible) economic cycle, to convert traded merchandise into merchandise wealth integrally.

Another way to interpret this impossibility is the following: if the economic cycle operates when in contact with only the merchandise reservoir at the economic temperature $T_{E,1}$ [U/€], receiving the traded merchandise $M_1$ [U] from that merchandise reservoir, and releasing the merchandise wealth $W_M=M_1$ [U] at an infinite economic temperature, this requires the traded merchandise transfer in the increasing economic temperature direction (in the decreasing unit price direction), which never happens in (natural) trading operations.

Considering now a cycle executed by the economic system when in contact with two merchandise reservoirs, the best (reversible) cycle is the economic Carnot cycle, but as seen in Section 4.2.5 even the economic Carnot cycle is unable to fully convert traded merchandise into merchandise wealth, as even it releases the merchandise wealth $\left|M_{2,C}\right|$ [U] given by Equation (46) to the merchandise reservoir at the lower economic temperature $T_{E,2}$ [U/€].

Any other economic reversible cycle executed by a system in contact with a higher number of merchandise reservoirs can be considered as composed of economic Carnot cycles, maintaining the impossibility of fully converting traded merchandise into merchandise wealth.

It should be noted that traded merchandise can be integrally converted into merchandise wealth in a process but not in a cycle, and the conclusions of this section are highlighted in its title as referring to a cycle. It is possible to purchase M [U] traded merchandise units at a given unit price and, after that, to detain these merchandise units M [U] as merchandise wealth, $W_M=M$ [U], out of the market, noting that this is not a cyclic process.

The bases of Thermodynamics were set based on pioneering studies on thermal engines. Once Thermodynamics is established as a discipline, it can be used to study the thermal engines. In the presented developments, the starting point was the economic Kelvin-Planck statement of the Second Law, setting the impossibility of fully converting traded merchandise into merchandise wealth in a cycle executed when in contact with a single merchandise reservoir. Once the economic Second Law was set and the merchandise economic entropy and its generation were defined, these developments were used in this section for a better understanding of even the starting point situations that allowed their establishment.

4.8.5. How Does the Economic Carnot Cycle Partially Convert Traded Merchandise in Merchandise Wealth?

Considering the economic Carnot cycle in Figure 6, the traded merchandise $M_1$ [U] is obtained (purchased) from the higher economic temperature $T_{E,1}$ [U/€] (at the lower unit price $p_1$ [€/U]), entering the economic system with the associated merchandise economic entropy transfer $M_1/T_{E,1}$ [€] (with the associated financial value transfer $M_1·p_1$ [€]). If no economic entropy generation (no financial value generation) exists in the reversible economic process, the traded merchandise $\left|M_2\right|$ [U] leaves the economic system (is sold) at the lower economic temperature $T_{E,2}$ [U/€] (at the higher unit price $p_2$ [€/U]), with the associated merchandise economic entropy transfer $\left|M_2\right|/T_{E,2}$ [€] (with the associated financial value transfer $\left|M_2\right|·p_2$ [€]), and from Equation (43) obeying

$${\displaystyle \frac{M_1}{T_{E,1}}}={\displaystyle \frac{\left|M_2\right|}{T_{E,2}}}\iff M_1·p_1=\left|M_2\right|·p_2[€]$$

(85)

Under these economically reversible conditions (null merchandise economic entropy generation), it suffices to sell the $\left|M_2\right|$ [U] merchandise units at the $p_2$ [€/U] higher unit price to obtain the financial value necessary to purchase the $M_1$ [U] merchandise units at the $p_1$ [€/U] lower unit price, and thus the wealth merchandise units $W_M=M_1-\left|M_2\right|$ [U] can be seen as obtained for free, at a null unit price, or at an infinite economic temperature. This is why they are merchandise wealth (which, if seen as traded merchandise, has a null unit price). This is the way how the economic Carnot cycle allows obtaining the merchandise wealth $W_M=M_1-\left|M_2\right|$ [U] from the traded merchandise received (entering the economic system) at the higher economic temperature $T_{E,1}$ [U/€] (received at the lower unit price $p_1$ [€/U]).

The normal economic activity is to create wealth and introduce it into the market, giving it non-null unit prices, and generating merchandise economic entropy (generating merchandise financial value). The Carnot cycle was conceived as the ideal cycle corresponding to the perfect (reversible) thermal engine to partially convert heat into mechanical work [1,2,3,18]. The economic normal is not having the perfect (reversible) economic engine partially converting traded merchandise into merchandise wealth. However, the economic Carnot cycle and its associated concepts are of major importance in the proposed and presented developments of the economic Second Law. The economic Carnot engine is also the best (perfect, reversible) to convert traded merchandise (received at a non-null unit price) into merchandise wealth (released at a null unit price), yet only partially.

The original and remarkable Nicolas Léonard Sadi Carnot’s work [18] is on the motive power of heat, heat moving in the decreasing temperature direction driven by a temperature difference, and the adequate engines that operate cyclically to deliver that motive power. Its proposed economic counterpart is the merchandise wealth-producing power of traded merchandise, traded merchandise moving in the decreasing economic temperature direction driven by an economic temperature difference (moving in the increasing merchandise unit price direction), and the adequate economic engines that operate cyclically to deliver that produced merchandise wealth.

4.8.6. Analyzing the Differential Equation Defining the Merchandise Economic Entropy

At this point, Equation (58), defining the merchandise economic entropy, can be revisited and better understood, what is made with the aid of Figure 14.

Under economic reversible (equilibrium) conditions, the traded merchandise $\delta M$ [U] enters the economic system crossing its boundary at the economic temperature $T_E$ [U/€], which is equal to the economic temperature of the system itself. The inlet of the traded merchandise $\delta M$ [U] into the economic system causes the change

$$d{N}_M=\delta M[\mathrm{U}]$$

(86)

on the merchandise units in the economic system. The merchandise economic entropy that enters the economic system with the traded merchandise $\delta M$ [U], through the economic system’s boundary, is $\delta M/T_E$ [€], and, using Equation (58), the change in the merchandise economic entropy of the economic system is thus

$$d{S}_{E,M}={\left({\displaystyle \frac{\delta M}{T_E}}\right)}_{rev}={\left({\displaystyle \frac{d{N}_M}{T_E}}\right)}_{rev}={\left(d{N}_M·p\right)}_{rev}[€]$$

(87)

As given by Equation (64), the merchandise economic entropy of the economic system is

$$S_{E,M}={\displaystyle \frac{N_M}{T_E}}=N_M·p[€]$$

(88)

and differentiating it gives

$$d{S}_{E,M}={\displaystyle \frac{d{N}_M}{T_E}}+N_M·d\left({\displaystyle \frac{1}{T_E}}\right)=d{N}_M·p+N_M·dp[€]$$

(89)

Under economically reversible (equilibrium) conditions, the merchandise economic temperature (the merchandise unit price) does not change to generate the merchandise economic entropy induced by that change. Thus, the terms in Equation (89) associated with the merchandise units change at the constant economic temperature (at the constant unit price) correspond to a reversible process, and the terms in Equation (89) associated with the merchandise economic temperature change (associated with the merchandise unit price change) at the constant number of merchandise units in the economic system correspond to an irreversible process, and Equation (89) can thus be written as

$$d{S}_{E,M}=\underset{rev}{\underbrace{{\displaystyle \frac{d{N}_M}{T_E}}}}+\underset{irr}{\underbrace{N_M·d\left({\displaystyle \frac{1}{T_E}}\right)}}=\underset{rev}{\underbrace{d{N}_M·p}}+\underset{irr}{\underbrace{N_M·dp}}[€]$$

(90)

It is thus clear why it is $d{S}_{E,M}={\left(\delta M/T_E\right)}_{rev}={\left(d{N}_M/T_E\right)}_{rev}={\left(d{N}_M·p\right)}_{rev}$ [€], as given by Equations (58) and (87), which, from the financial viewpoint, is an obvious result. Changing the unit price (the economic temperature) of traded merchandise with no other effect corresponds to generate merchandise economic entropy, corresponding thus to an irreversible process.

A similar interpretation and clarification in Thermodynamics [1,2,3] is not allowed, as there is no physical meaning for the inverse of the absolute temperature. This is just one example of how the present developments, based on what is made in Thermodynamics to look at Economics, may help a better and full understanding of what happens in Thermodynamics. As referred to in Section 1, the present work ‘allows alternative teaching and learning approaches, the more easily understandable of one helping to understand the other’.

5. The Economic Third Law

Once the Zeroth, First, and Second Laws of Thermodynamics are set, the Third Law sets that is null the entropy of a thermodynamic system at the null absolute temperature [1,2,3]. Similarly, once the economic Zeroth, First, and Second Laws are set, the economic Third Law sets that is null the merchandise economic entropy of an economic system at the null economic temperature (sets that is null the merchandise financial value of an economic system whose merchandise units have an infinite unit price).

Consider the simplest economic system with no merchandise unit generation or destruction and no monetary units. The economic system is initially at state 0, and its economic temperature is lowered toward the null economic temperature through a sequence of economically isothermal (constant unit price) and economically adiabatic (no traded merchandise transfer) reversible processes, as illustrated in Figure 15. This sequence of processes is limited between the edges presented by the thicker lines in Figure 15.

During each of the economically adiabatic processes (horizontal processes), the economic system releases some merchandise wealth and economically colds down (the economic temperature decreases, and correspondingly the unit price increases), and during each of the economically isothermal processes (vertical processes) the system releases some traded merchandise units. When dealing with thermodynamic systems, closed relationships exist between internal energy, temperature, and entropy [1,2,3]. This is not the case when dealing with economic systems. As mentioned in Section 2.1 and Section 4.6.3, the economic temperature is not exactly a property of the economic system, and it is relevant as it affects the merchandise economic entropy entering or leaving the economic system (the unit price of the merchandise entering and/or leaving the economic system) through its boundary.

In Thermodynamics, isothermal compression or magnetization processes are used [3]. However, this is not the case for the economic system. In this case, it may be assumed that the economic system is allowed to be in economic contact with a merchandise reservoir at a given economic temperature and releases some traded merchandise at the unit price corresponding to that economic temperature (vertical processes in Figure 15). After that process, some action exists in the economic system, and it releases merchandise wealth (releases merchandise at null unit price, corresponding to donating those merchandise units) in an economically adiabatic (no traded merchandise transfer) process (horizontal processes in Figure 15). Such an economically adiabatic release of some merchandise wealth means a decrease in the number of merchandise units in the economic system, and if its merchandise economic entropy (its merchandise financial value) remains constant (horizontal lines in Figure 15), it forcedly happens that the unit price of the merchandise units remaining in the economic system increases, and their economic temperature correspondingly decreases. After that, some traded merchandise units are sold at a given unit price (which corresponds to a given economic temperature), and a situation is reached such that no more traded merchandise is sold at that unit price. After that, new potential purchasers are needed to purchase more merchandise units at a higher unit price (at a lower economic temperature). This sequence of processes is repeated until no more traded merchandise units are sold.

From Equation (58) $d{S}_{E,M}={\left(\delta M/T_E\right)}_{rev}$ [€], and for each of the economic isothermal processes in Figure 15 as $\Delta S_{E,M}<0$ [€] it is $M<0$ [U] (traded merchandise units released by the economic system). The merchandise units balance Equation (13) for the economically adiabatic expansion gives $d{N}_M=-\delta W_M<0$ [U], and a decrease in the number of merchandise units in the economic system leads to a decrease in the economic temperature (a lower number of merchandise units at the same merchandise economic entropy means a higher unit price, that is, a lower economic temperature). This decrease in economic temperature is followed by an economically isothermal decrease in merchandise entropy. This pair of processes, in that order, is successively repeated, as illustrated in Figure 15.

As can be seen from Figure 15 it is

$$\underset{T_E\to 0}{\lim }={\left(-\Delta S_{E,M}\right)}_{T_E}=0[€]$$

(91)

where ${\left(-\Delta S_{E,M}\right)}_{T_E}$ [€] is the merchandise economic entropy change (the symmetric of the merchandise economic entropy decrease) in each of the economically isothermal reversible (vertical) processes. As the considered thicker edges in Figure 15 intercept at $T_E=0$ [U/€], it can be stated that

$$S_{E,M}\left(T_E\to 0\right)=0[€]$$

(92)

which is the analytical statement of the economic Third Law. This result can be interpreted as a setting that an economic system at null economic temperature has null merchandise economic entropy. Given the meaning of the economic temperature (Equation (1)), Equation (92) can be rewritten as

$$S_{E,M}\left(p\to \infty \right)=0[€]$$

(93)

In other words, an economic system whose merchandise units have an infinite unit price has null merchandise financial value.

The considered successively traded merchandise releases during the economically isothermal reversible processes (vertical processes in Figure 15), followed by the reversible merchandise wealth release during the economically adiabatic process (horizontal processes in Figure 15), similar to a donation (a merchandise release at null unit price), correspond, respectively, to successive merchandise units and merchandise economic entropy (merchandise financial value) drains from the economic system, accompanied by successive economic temperature decreases (accompanied by successive unit price increases) during the economically adiabatic merchandise donation processes.

Equation (58) defines the merchandise economic entropy through a differential expression, but it does not set the zero, or absolute value, of the merchandise economic entropy. The economic Third Law, expressed through Equations (92) or (93), allows defining that absolute value. This is how the economic Second and Third Laws are related.

The result in Equation (93) can be understood from an economic viewpoint by looking at the merchandise demand curve in Figure 16 [4,5].

As the merchandise unit price increases, the number of traded merchandise units decreases. The number of traded merchandise units is null at the limit of an infinite merchandise unit price. In this sense, the financial value of a merchandise whose unit price is infinite is null. Even if with an infinite unit price, that merchandise would not be traded.

6. Conclusions

As referred to in the Introduction, this work proposes a Four Laws structure, analogous to the Thermodynamics Four Laws structure, for looking at Economics through the eyes of Thermodynamics. The main concepts, definitions, variables, and equations involved in the developments itemized in the Introduction were fully achieved.

The present work complements the work in [6], as it includes the statement of the economic Zeroth Law, the developments that, starting from the base economic statements of the Second Law resulting from economic observations, lead to the definition of economic entropy and economic entropy generation, and the setting of the economic Third Law. In addition to the proposed Four Laws structure, the presented developments introduce and explore a large set of concepts, variables, and relations that are of major relevance for looking at Economics through the eyes of Thermodynamics.

Economics and Thermodynamics may benefit from the proposed Four Laws structure, including the large set of concepts, variables, and relations included in the presented developments, from both scientific and pedagogical viewpoints.