Von Neumann – A Chemist’s Approach to Economics

Katalin Martinás

1                       Introduction

Neumann, in his 1937 paper introduced several important concepts (Zalai, 1999, 2003), in spite of the fact that he, as a mathematician, chemist (and physicist), did not deal systematically with economics. The History of Economic Thought Website summarizes the contributions of his model to economic theory as follows: “Von Neumann introduced:(1) linear system of repro­duction, (2) the concept of “activity analysis” production sets, (3) price-cost and demand-supply inequalities, (4) firstly (the generalization) of Brouwer’s fixed-point theorem to prove the existence of equilibrium, later to be known as Kakutani’s Fixed Point Theorem; (5) the minimax and maximin solution methods and the saddle-point characterizations; (6) early statements of duality theorems of mathematical programming and complementary slackness conditions; (7) a novel manner of incorporating fixed and circulating capital via joint production; and (8) the derivation of the “Golden Rule” – showing that the rate of interest is related to the rate of growth.”

On the other hand it has repeatedly been maintained that the model is not a “very good economics” (Koopmans, 1974). Heinz D. Kurz and Neri Salvadori argue that the criticism is due to the close link with the classical spirit. Nevertheless, there is no evidence that von Neumann dealt systematically with economics, and he studied the works of economists. We argue that the features of the von Neumann model which are considered as classical elements in spirit come from the fact that the physical parts of activity, which will be important for a chemist, were also important for the classical economists, and they disappeared from economics only via the neoclassical revolution.

In this paper we wish to illustrate that Neumann’s economic model is in close relation with the thermodynamic description of chemical systems. It is not a copy, but a transformation to the special problem of equilibrium growth. It is not the application of thermodynamic results or formula but it is the methodology of thermodynamics. That interpretation explains Neumann’s statement: “It is clear which type of theoretical model belongs to our assumptions”. For the proof first we give a summary on the methodology of thermodynamics, and then the structure of a thermodynamic-like (irreversible) economic theory is outlined. The Neumann model will be derived as the restriction of the general theory for Neumann economies.

The novelties (1), (2) and (7) have chemical counterparts, furthermore the duality and the use of inequalities have some links to thermodynamics. András Bródy analyzed the duality question. The role of inequalities in thermodynamics would be the subject of a lecture. Here we investigate the basic similarity between chemistry and economics, namely that they are concerned with transformation and transport of material. Formal description of the material flows in economics and in chemistry lead to equations of the same structures, with some important differences. The Neumann model is not a copy of thermodynamic results to economics, but an economic theory which utilizes the results of thermodynamic approach (the philosophy and logic).


1.1                                      Arguments for thermodynamic interpretations

1. Von Neumann did not reveal the origins of his model. This fact may explain the puzzling remark of von Neumann: “It is obvious to what kind of theoretical models the above assumptions correspond” (1945, p2). As thermodynamics is the “language of chemistry”, the thermodynamic argumentation was obvious for Neumann.

2. Von Neumann himself has called attention to an interesting formal analogy that exists between the economic phenomena and thermodynamics. Namely, the role of the profit function “appears to be similar to that of thermodynamic potentials”, Bródy (1989) revisited von Neumann’s conjecture and has offered a potential explanation of its deeper meaning. He discussed the formal analogy between the minimum and maximum principles of thermo­dynamics and the minimax theorem. Further, he pointed out the duality of thermodynamics (extensive and intensive parameters) and the duality principle of prices and the volume of production.

3. “The similarity will persist in its full phenomenological generality (independently from our restrictive idealizations)” (1945, p1).

This is the key to our argument. First we formulate the First and Second Law of Economics. First law states that the economic activity can be described via the changes of goods. Second Law states the directionality of economic processes. (Later we give a more precise definition.) The First Law, in its direct or indirect form, was already articulated by economists. The classical economists accepted it. There are a lot of economic works and theories, which were founded on the First Law, see for instance Sraffa (1960), the input-output model of Leontief (1954), and Kornai (1984). On the other hand, the neoclassical approach considers that aspect as trivial and non-important. Nevertheless, they did not incorporate the Second Law. So this thermodynamic-like approach is not a classical nor is it a neoclassical approach. From the First and Second Law of economics the derivation of the Neumann model is straightforward.

Organization of the paper is as follows. In Section 2 the thermodynamic description of chemical systems will be outlined. First and Second Law will be given in such form, which can be applied also in economics. In Section 3 a thermodynamic (or irreversible) microeconomics is outlined, with evolution equation for a Neumann-type economy. The numerical solutions of the model economy show the theoretical results. The golden rule appears as a singularity on the plot lifetime of the economy versus interest rate, indicating that there is only one interest rate which leads to stable growth. In the fourth part the basic novelties of the Neumann model are derived from thermodynamic considerations.


2                       Thermodynamics

2.1                                     A quick summary of the history of economics and thermodynamics

Analogous properties of chemical and economic processes are well known for a long time. Kalman Szily (a physicist) gave a lecture in 1871 with the title Communistic state of the physical world, where he used some economic examples to explain the First and Second Law of Thermodynamics. In 1903, just 100 year ago, when von Neumann was born, Jankovich [Jankovich 1903] published a paper on the mechanical foundation of value. Imre Fenyes (professor of thermodynamics) emphasized the similarity between the economic goods and the thermodynamic quantities. Micro­economics and thermodynamics are both based on the idea of exchange. In thermodynamics the irreversibility of exchanges is a key idea and one that is sometimes difficult for my physics students to understand. So about 15 years ago I went looking for examples of irreversibility in economic theory that I could use in my teaching of thermodynamics. What I found, however, was no irreversibility in the neoclassical paradigm. As a physicist this struck me as preposterous and incredible. Without irreversibility, microeconomics might be a wonderful mathematical theory, but it could not offer a theory of economic activity. This encounter marked the beginning of my long-term interest in economic research. In 1986 with András Bródy and Konstantin Sajó [Bródy et al., 1986, 1994] I wrote a paper based on the irreversible properties of exchanges, where we showed the existence of an „economic entropy function for neoclassical economics. Later it became clear that this paper was a mistake. A lot of concepts were used simultaneously with different thermodynamic and economic interpretations. Thermodynamic interpretation did not coincide with the neoclassical interpretation.

Neoclassical economics is basically a static theory without irreversibility [Samuelson, 1983]. The structure of neoclassical economic theory and that of thermodynamics are distinctly different. Later, with Robert U. Ayres, I started to elaborate a non-neoclassical economic theory based on the framework suggested by the irreversible thermodynamic theory. When it was finished, it was easy to show that this approach reduces to the Neumann model in the case of a Neumann economy. On the other hand the equilibrium approach of neoclassical economics to economic processes kills the possibility of the incorporation of irreversibility of economic processes, as well as the material side of economic actions. The Neumann model is not in the neoclassical trend.


2.2                                     Laws of Thermodynamics

There are two basic approaches to understand the equilibra­tion process: a macro (ther­mo­dy­na­mics) and a micro (statistical mechanics) approach. Nowadays the statistical approach is more popular. Statistical mechanics, developed initially by Ludwig Boltzmann, defines entropy as the logarithm of the number of “micro­states”. Boltzmann’s definition of entropy rather suggests the order/disorder metaphor. Boltzmann’s statistical approach is a useful tool in a wide range of applications.

Classical (or phenomenological) thermodynamics starts from the observed properties of our material world. There are construc­tions starting from the relations of heat and work [Clausius 1865; Landsberg 1961]. They can lead only to analogies, as there is no heat and work in economics. But there are constructions too, which start from the general properties of systems, where heat and work are derived concepts [Tisza 1966; Callen 1960; Martinas 1992]. These approaches do not use concepts directly related to physical phenomena, they can be applied for other (non-physical) systems.

In the traditional approach the three laws of thermodynamics are: The Zeroth Law, which expresses the transitivity of equilibrium, and which is often said to imply the existence of temperature as a parameterization of equilibrium states. The First Law, which states the conservation of energy. Three popular formulations of The Second Law are:

Clausius: No process is possible which results solely in transferring heat to a hotter body.

Kelvin (and Planck): No process is possible which results solely that a body is cooled and work is done.

Carathéodory: In any neighborhood of any state there are states that cannot be reached from it by an adiabatic process.

Modern non-equilibrium ther­mo­dy­na­mics began with the “local equilibri­um” hypothesis introduced by Max Planck [Planck 1892]. The foundations of irreversible ther­mo­dy­na­mics were laid down by Lars Onsager in 1931 [Onsager 1931a, b]. The real success and acceptance of non-equilibrium thermody­namics is due to the Brussels school directed by Ilya Prigogine. Onsager finally received a Nobel Prize in chemistry in 1968, while Prigogine was similarly rewarded for his work in 1977. There are also several approaches to extend the realm of ther­mo­dy­na­mics beyond the local equilibrium. The two main groups are “rational ther­mo­dy­na­mics” [Truesdel 1969], and “extended ther­mo­dy­na­mics” [Jou et al 1992]).


2.2.1 First Law

There were many attempts to introduce the First Law of economics, looking for a quantity similar to energy. But, as there is no economic quantity having such a conservation law as energy, these attempts were unsuccessful. The real importance of energy conservation law for thermo­dynamics is not the conservation, but the existence of energy. It was needed for the quantitative description of physical systems.

In ther­mo­dy­na­mic investigations it is worthwhile to distinguish between the extensive vari­ables (volume, energy) and intensive variables (e.g. temperature, pressure). An extensive variable must satisfy two conditions. First, its time dependence is described by the generic balance equation:

dX/dt = J+G                                                                                             (2.1)

where J is the flux, and G is a generalized source term. Second, it must be additive, in the sense that if Xa and Xb are values of the variables for two systems ‘a’ and ‘b’, the variable has the value Xa+Xb for the combined system, consisting of the union of the two. A general rule of ther­mo­dy­na­mics states that all the interactions of the ther­mo­dy­na­mic system with its environment take place through the flows of extensive variables. Mass, energy, the number of molecules and total volume are examples of extensive properties. That property is summarized in the First Law.

First Law. Any simple system has particular states that are characterized completely by extensive quantities.

The postulate reflects an important feature of thermodynamic systems. The evolution of the system is governed by the balance equations of extensive quantities, in the form:

dXki/dt =ĺkl,i Jkl,i+Gki                                                                                (2.2)

where index i and l identify the subsystem, Xik is the quantity of the extensive parameter k, Jil,k denotes the flow from the system l to the system i.

The independent set of extensive variables necessary to describe a given system is determined essentially by trial and error. The choice is not unique. It depends on one’s purpose or (from another perspective) on the accuracy of one’s measurements. For example, consider the air in a room. For certain purposes it is sufficient to treat it as an equilibrium gas. But for a more precise measurement one has to take all the different types of molecules present in the air into account. In still more precise calculations one might also consider the different isotopes. Going to extremely fine details it might be necessary to consider the internal structures of the atoms. In real calculations it is necessary that we take into account only those details what are important for the problem in question. This criterion does not mean that the selection of variables is arbitrary. Thermodynamics provides a rule for self-consistency.


2.2.2 Chemical reactions

As chemical reaction formalism and methodology has a special role in understanding of the chemical roots of the Neumann model, we summarize the most important elements.

For conserved quantities (energy, mass, number of different atoms) the source term is zero. In chemistry the molecules change. For instance, when propane gas, C3H8 is burned in oxygen, the products are carbon dioxide and water. This can be written as a chemical equation:

C3H8 + O2 ® CO2 + H2O                                                                         (2.3)

If chemistry were not a quantitative science, then this description of the reaction, identifying both the inputs and the outputs, would be adequate. But we expect more, and the nature tells us more. In the reaction the quantity of different atoms does not change (conservation law!). We must add numerical coefficients that tell how many of each kind of molecule are involved, and then there will be the same number of each kind of atom on the left and right sides of the equation, since atoms are neither created nor destroyed in a chemical reaction. The correct coefficients for the four substances in the propane burning equation are 1, 5, 3, and 4:

C3H8 +5 O2 ® 3CO2 + 4H2O                                                                   (2.4)

Each side of this balanced equation contains 3 carbon atoms, 8 hydrogen atoms, and 10 oxygen atoms. The coefficients are called stoichiometric coefficients. A balanced chemical equation describes the starting materials and products. It also tells us that the number of each kind of atom entering the reaction is the same as the number of those leaving it. Each type of atom individually is conserved during the reaction. What a balanced equation does not tell us is the molecular mechanism. Balanced equations are used to calculate the expected yield (quantity of product) from a reaction.

Chemists introduced a concept, which will appear also in Neumann’s work as the intensity of the operation of the ith process (qi). In chemistry the extent (or advancement ) of a chemical reaction is measured as the change in the number of moles of the substance whose stoichiometric coefficient is unity. Formation of water from hydrogen and oxygen by reaction

2H2 + O2 ® 2 H2O                                                                                   (2.5)

meaning that 2 moles of H2 and 1 mole of oxygen react together to give 2 moles of water. In chemistry the mole numbers needed for the reaction are called stoichiometric coefficients. With Neumann’s notation they form a line of the input and output matrices:

The input matrix: A = (2,1,0)

The output matrix B = (0,0,2)

If the intensity (extent) of the reaction is q, then the change of the relevant quantities: the quantity of water nH2O, quantity of oxygen and hydrogen (nH2 and nO2) are:

dnH2O= 2 q,

dnH2 = - 2q,                                                                                               (2.6)

dnO2 = - q,

or in general form:

dXi/dt = Ji + Aq – Bq                                                                               (2.7)


2.2.3 Second Law

The Second Law of thermodynamics is essentially different from the First Law, since it deals with the direction in which a process takes place in nature. It expresses the preferences of Nature. Not every change which is consistent with the balance equations is a possible change. The conservation laws do not suffice for a unique determination of natural processes. As for instance, in the previous example, the water equation offers no information, whether hydrogen and oxygen actually combine to form water, or water decomposes into hydrogen and oxygen or whether such a process can go into both directions.

The essence of the Second law is that all the independent elementary (infinitesimal) processes that might take place may be divided into three types: natural processes, unnatural (forbidden) processes, and reversible processes:

·        Natural processes are all such that actually do occur. (Example: heavy body falls down.)

·        Unnatural process (the reverse of a natural process): such a process never occurs.

·        As a limiting case between natural and unnatural processes are the reversible processes. They do not actually occur.

The Second Law implies a relation between the quantities connected with the initial and final states of any natural process. The final state of a natural process has to be discriminated from the initial state, while in a reversible process they have to be in some sense equivalent. Entropy does it. If there is no heat exchange, then in a natural (irreversible) process the entropy is increasing, in a reversible process the entropy is constant.

The entropy principle contains the direction of natural processes; it establishes the basis to derive the thermodynamic force law.


2.2.4. Non-equilibrium thermodynamics – Dynamics

The balance equations describe the time evolution of the systems, the relation of flows and the distribution of stocks has to be defined empirically. These relations are empirical, material dependent, but not arbitrary. The flows must obey the Second Law. That condition has a great power. It allows us to introduce the concept of thermodynamic forces, and the force law which connects the flows with the state variables.

The uni-directionality of natural processes is formulated in the form of a dynamic law in non-equilibrium thermodynamics. The change (flow) is proportional to the force (difference in intensive parameters, dYk)

Ji = ĺ Lik dYk                                                                                                                            (2.8)

where L is the so called conductivity matrix. It is positive definite. That property of matrix L follows from the Second Law, as it expresses the uni-directionality of spontaneous changes, or “time’s arrow”. That law is experimentally verified. Historically they were first formulated as empirical laws (Ohm’s law, Fourier’s law, etc.).


3                       Thermodynamic based microeconomics

The formal analogy between neoclassical economic theory and thermodynamics seemed to be promising. There are a lot of works, which aimed to find economic counterparts of the relevant thermodynamic quantities (entropy, energy, temperature, chemical potential). The usual interpretation is that utility is similar to entropy, money is similar to energy. Nevertheless, these analogies are of limited validity (Martinas, 2001). Utility does not have the property needed to be similar to entropy. Further, the structure of the two theorems is different. Neoclassical economics is a timeless approach, while thermodynamics is based on the time arrow. The explanation was given in an excellent summary on the use of physical analogies by early neoclassical economists (Mirowski, 1984). Mirowski writes: “The metaphor of energy/utility which neoclassical economics appropriated was derived from the physics of a specific historical period: the years of the mid-nineteenth century just prior to the elaboration of the second law of thermodynamics.” Yet pre-entropic physics has been basically a theory based on mechanical considerations, without a “time-arrow”, without irreversibility. As the concepts of modern economics were developed from this pre-entropic formalism, it is natural that modern economics is incompatible with thermodynamics. But it meant a challenge to build a new economic theory, which exploits the uni-directionality of economic processes. In this chapter we outline the structure of irreversible economic theory. The numerical solutions of dynamic equations for a Neumann economy confirm the golden rule. There is only one interest rate, which allows stable exponential growth. All the other values allow growth with limited stability. That is our first proof for the thermodynamic roots of the Neumann model. The second proof will be in the next section where the basic inequalities are derived from the general equations.


3.1                                     Irreversible economics

For the mathematical structure of irreversible microeconomics see Martinas 1995, and Ayres and Martinas 2003. Here we summarize the most important definitions and concepts.

Definition 1: An economic agent (EA) is defined for our purposes as the smallest entity with an implicit or explicit de­ci­sion-making rule. An EA would normally be either a firm or an individual. EA are characterized by their scope of activities, by their know­ledge, experiences and by their stocks. Our primary interest is the change of stocks and its economic effect. Every stock which can be affected by the economic acti­vity of an agent can be listed, and those also, which affect the economic activity of the agent. The list of stocks may contain the money, but it is not necessary.

For an individual actor the processes which result in stock changes can be grouped as follows. In consumption the stock change is always negative. Production is a transformation of stocks from the initial form to a final form. Here the change is positive for the products and byproducts, and negative for the input materials. Total stock increases only when there is an input from the nature. Otherwise the obligatory losses decrease the total stock.

Third type is the trade. In this case the total change of stocks is near to zero, as it changes only the position (and owner) of stocks. Further, there is group called societal effects: tax payment, robbery, economic aid are examples for further stock changes. Here the general rule is that it is positive only below a certain limit value of stocks. Society helps the poor and those who are in crises. For a normal agent the effect is negative. At last, but not least the physical (natural) changes. Second Law of thermodynamics always decreases the stocks. It appears in the form of degradation, depreciation, losses, and damages.

Based on the previously mentioned decisions concerning the goods, these decisions can be divided into two groups; one of these is economic decision – trade and production. The other group consists of non-economic decisions: all the decisions not belonging to the first group, for instance tax payment, robbery, luxury consumption. It is summarized in the following definition:

Definition 2: Economic actions are trade and production.

Trade and production are des­cribed as de­ci­sions. EA selects or rejects the offers provided by other EAs or (in case of production) by his/her internal state. We call a decision trade when the economic agent selects the flow, Ji. Production-consumption decision selects the pro­duc­tion.

Decisions are tantamount to selections from a limited set of possibilities for immediate action. The set of possibilities is constrained by the external environment (for example, the legal framework) and by the assets of the agent, including financial assets, physical assets and intangibles such as knowledge, know-how, reputation, and so on.

Proposition 1 (First Law): Evolution of an economic system is described by the balance equation for stocks of goods and money.

(The balance equations are derived and discussed in Appendix 1). First Law in economics seems to be a natural approach as the stocks, resources, goods and money are extensive variables, as they are additive and there is a balance equation defining their changes.

The real question is whether it is an important side of economic activity, which is described by the stock changes. Neoclassical economics considers that aspect as trivial, non-important. From this basis the answer is no. If one accepts neoclassical economics as a full description of economic activity, then the thermodynamic approach is meaningless. The thermodynamic-like approach is a non-neoclassical approach. That materialist orientation of classical economics was expressed by Petty (cited by Kurz [2003] as „ The Method I take to do this, is not yet very usual; for instead of using only comparative and superlative Words, and intellectual Arguments, I have taken the course (as a Specimen of the Political Arithmetick I have long aimed at) to express myself in Terms of Number, Weight or Measure; to use only Arguments of Sense, and to consider only such Causes, as have visible foundations in Nature; leaving those that depend upon the mutable Minds, Opinions, Appetites and Passions of particular Men, to the Consideration of others ... (Petty, [1899] )”

Proposition 2 (Second Law of Economics): No agent makes an economic decision to decrease the expected welfare

Second Law in economics: the uni-directionality of economic processes was also formulated (but without the First law). The no-loss rule first appeared in the Austrian school when Menger stated the necessary conditions for an exchange [Menger 1871]. For a free exchange of goods among economizing individuals the following triad of conditions must be fulfilled:

a.       one economizing individual must have command of goods which have a smaller value to him/her then other quantities of goods at the disposal of another economizing individual who evaluates the goods in reverse fashion,

b.      the two economizing individuals must have recognized this relationship, and

c.       they must have the power actually to perform the exchange of goods.

An essential prerequisite for the exchange is missing if any of the above three conditions is not present. The first condition, essential in free economic exchange, is the no-loss rule: an economic individual never acts if that action would result in an immediate loss. The no-loss rule is postulated as a decision rule for economic agents instead of utility/profit maximization. It is extremely important to note that the no-loss rule holds only for the moment of action. The natural and economic environments as well as the economic agents themselves are in continuous transition, so that yesterday’s decision may seem today to have been a bad decision. The no-loss criterion is weaker than the utility maximum principle. It presupposes only that every economic unit has common sense, and hence does not do anything which impairs its economic state. It does not presume perfect “rationality”, that is, it does not suppose perfect foresight, nor does it necessarily follow that the actions taken are optimal.


3.2                                     Economic welfare function

No loss rule requires that the economic welfare of an economic agent is a function of the stocks of goods and money belonging to the economic agent: Z = Z(X) The proof is given in [Martinás, 1996]. The equation Z = const defines an N –1 dimensional surface consisting of the indifference points. The sign convention is selected so that dZ> 0 for allowed (no-loss) processes, and dZ < 0 for forbidden (loss-making) transactions. No-loss rule states that for economic processes (trade and production) economic welfare is not expected to decrease, whence for all ‘allowable’ economic transactions the economic welfare is increasing. As the non-economic processes may decrease the economic welfare, the real change is not predefined. Consumption beyond subsistence requirements is one way to decrease welfare. Theft and disaster losses are also possible, and they always decrease economic welfare. Properties of the economic welfare function are as follows [Ayres and Martinás 1996]:

(i)                  Since economic welfare is a positive attribute, a function that measures economic welfare must be non-negative. Normally in trade and production processes dZ > 0 .

(ii)                Economic welfare comprises all goods and money, or money-equivalents (like receivables) that are owned outright (net of mortgages, debts or other encumbrances). The terms “own”, “owned”, “ownership” etc. are shorthand for a more cumbersome phrase, such as “to which the economic agent has enforceable exclusive access”.

(iii)               An increase in the agent’s ownership of stocks of beneficial goods or money results, ceteris paribus, in an increase in the agent’s economic welfare.

(iv)              There are cases, when doubling all stocks will double economic welfare. This implies that the economic welfare function should have the property of homogeneity in the first degree. This is a useful property when it comes to selecting representative mathematical forms for Z.

(v)                The Z-function implicitly contains all information on the expectations of the agent. Its partial derivatives are the subjective value functions.

Assuming the function Z is continuous and differentiable, the partial derivatives in respect of the stocks can be interpreted as the marginal Z-value of the good i. It is measured in welfare/quantity units. Similarly wM = Z/M is the marginal Z-value of money, and vi = Z/Xi /Z/M is the marginal value of the good i, measured in monetary units. With the above notation the differential change of economic welfare becomes

dZ = wM (ĺ vidXi + dM)                                                                           (3.1)

The expected gain (profit) in trade of a unit of good i for price pi is F = wM(vi – pi), the expected gain for production F = vB – vA. For the following analysis it is more convenient to introduce a production matrix in the form C = B - A, then F = vC. The form of the force law is based on empirical data, as a first approximation we apply the linear law, then

Jn = Ln (vB vA)                                                                                      (3.2)

and the traded quantity when agent k trades with the agent m at price pi will be

Jkm,i = Lkm,i (vi – pi)                                                                                   (3.3)

Trade is viable only if the agent m agrees the same quantity with opposite sign. So the above relation gives an equation for pair-wise exchanges. The agreed price refers to supply-demand equilibrium. The no-loss principle demands the positive definiteness of matrix L, so the price vector is uniquely defined.

The equation system is closed. (We omit the indices for the sake of clarity and brevity.)

Balance equations:

dX/dt = L(v-p) + Ln(vC)C                                                                        (3.4)

v= Z/X (Z/M)-1 =v(X)                                                                       (3.5)

Price, p comes form the solution of equation

ĺ Jnm,i =ĺ Lnm,i (vi – pi )= 0                                                                      (3.6)

where the summation is for the agents participating in the bargaining process.

A numerical solution needs the following data: identification of the agents, their activity, initial stocks, welfare function and reaction parameters plus a description of the learning and adaptation process (feedback loops for adjustment of L parameters). Technological innovations and monetary policy of the economic system may also have to be specified. Further, exogenous effects such as consumption not connected with production, taxation, depreciation, and natural constraints, if any, must be specified.


4                       Neumann economy

Our minimum model of an economy has 3 economic agents, corresponding to sectors, namely: agriculture, industry, and households and we apply the simplifying assumptions of Neumann, namely:

·        Matrix A and B are constant (they do not depend on the production intensity).

·        Nature is infinite (there are free goods of nature in quantities without limit).

·        Consumption is proportional to the production.

·        Labor is considered as a normal stock (the households produce it).

Differences between irreversible economic approach and the Neumann model:

·        Fixed price and perfect trade assumptions are released. The agents decide the production intensity, and by the bargaining rule they agree in the prices and traded quantities.

·        Consumption contains a fixed, production independent part.

·        Agents get interest payment for their money stock.


4.1                                     Selection of the welfare function

Results of simulations show that the global properties of solutions are not sensitive to the form of the Z-function, although the minor details (‘fine structure’) naturally are affected by it. We selected the logarithmic form:

Zn =ĺi Xin log (CnMn/Xin)

where Xin is the stock i of the agent n, Mn is the money stock of the agent n, Cn is constant.

Initial stocks were selected as:

Table1: Initial Stock vectors






















Input –output matrix:

To reflect the need to compensate for excess consumption and depreciation losses we have specified minimum subsistence production levels for each agent. For agriculture, this could be interpreted as the output needed to provide food, feed and seed for the farmers and their livestock. For the industry sector, it is the requirement to compensate for depreciation. For households, there is a minimum food and tool requirement, to produce labor. These minimum requirements would normally be expressed as vectors that must be subtracted from the final stocks for each agent after each cycle. As a practical simplification, with mathematically equivalent effect, we have incorporated non-zero diagonal (own-consumption) terms in the input-output matrix, as shown below in Table 3-4.

Table 2: Output matrix


















Table 3: Input matrix


















The parameter L for all trades is assumed to be unity, viz. L=1. Each agent trades with each other agent. The production decision parameters are: L1 = 0.352, L2= 0.288 and L3 = 0.352. The interest rate is given for the money. After each cycle the money of the agents is multiplied with a.


4.2                                     Numerical solutions

The numerical solution of dynamic equation establishes the “real time” behavior of the Neumann economy. First the values are calculated, the initial values are shown in Table 4:


Table 4: Initial Values































The agents agree in price, with the rule

pi = (v1i+v2i+v3i)/3                                                                                      (4.2)

The results will be 2,5 and 10. The new stocks after the trade will be given by the rule X’ = X + (v-p). The money of the agents changes as M’ = M – (v-p)p. The agents calculate the new values (defined by the new stocks). They calculate the potential welfare increase (Av-Bv), and the production intensity (x= L(Av-Bv)). The stocks are modified, and the compulsory consumption is done. In the next step the money is increased by the interest. That is a complete cycle. For the characterization of the system we selected the total intensity of production. In Figure 1 the intensity of production is plotted as a function of time (number of completed cycles) for a= 1.0005. That interest rate is not the equilibrium one. On the curve 3 parts has to be distinguished. The initial period (in a magnified form it is on Figure 2) is the non-equilibrium part, as the initial stock distribution did not correspond to the equilibrium. The initial time period is random, but later the order develops from chaos. The interaction of agents (in the trade) leads to harmonization. The fluctuations decrease, instead of the random changes quasi-periodic oscillations emerge with decreasing amplitudes, finally the system finds the equilibrium path. There is a near-exponential growth of the production intensity. Nevertheless the agents’ welfare increases unequally. This inequality leads to the loss of stability.

In Figure 3 the lifetime is plotted as a function of interest rate. It confirms the Golden rule. There is only one interest rate, which has infinite lifetime. All other interest rate leads to finite stability.

The figures confirm Neumann’s result, the golden result, for a given technology there is only one interest rate, which gives a stable growth. All the other interest rates give growth with limited stability, but in the nearly stable growth regimes the price ratio is nearly constant.


5                       Neumann model and thermodynamics

The Neumann model is a simplified version of the irreversible economic model, as it follows from the basic equations, taking into account the simplifications due to the use of Neumann economy.

The First Law in general form is

dX/dt = J + (B-A)x                                                                                   (5.1)

As the aim is to investigate the equilibrium growth problem, the following simplifying assumptions can be applied:

1.      There is perfect trade, with fixed prices p and zero price for the trade with nature.

2.      The effect of stock changes are neglected, that is dX/dt = 0.

It implies that nature is considered as infinite reservoir and the equilibrium growth condition is

ĺj bij·xj =(1+lĺj aij·xj,     i = 1, 2, Ľ , n                                                 (5.2)

For a non-equilibrium growth

ĺj bij·xj ł (1+lĺj aij·xj,   i = 1, 2, Ľ , n                                                 (5.3)

The Second Law in the Neumann model can be traced back. In irreversible economic approach the necessary condition for production is

ĺi pi·bij - ·ĺi vij·aij > 0 for real                                                                 (5.4)

ĺi pi·bij - ·ĺi vij·aij < 0 for forbidden                                                        (5.5)

where vij is for the value. In the equilibrium approach (with fixed prices in the production decision) the prices enter to the place of values, so the necessary condition for production is

ĺi pi·bij - ·ĺi pi·aij > 0 for real                                                                  (5.6)

ĺi pi·bij - ·ĺi pi·aij < 0 for forbidden                                                         (5.7)

The above criteria are valid for agents who are taking part only in production and trade processes. If there is a bank, who gives interest payment for the money, then the reference condition for the no loss estimation will be different. With the thermodynamic analogy, we have to look for the relevant potential. In irreversible economic approach the necessary condition for production is

ĺi pi·bij - (1+pĺi pi·aij > 0 for real                                                         (5.8)

ĺi pi·bij - (1+pĺi pi·aij < 0 for forbidden                                               (5.9)

That uni-directionality of the process is expressed by the no loss formulated by Neumann as “the activities that do not yield the maximum rate of return, will not be used in equilibrium” (the Rule of Idle Activities).


6                        Conclusions

The Neumann model is not a neoclassical approach in economics; it has a lot of common with classical economics. Here we collected the arguments, that Neumann model is really a thermodynamic based approach to economics. In one hand, as a chemist Neumann knew thermodynamics. On the other hand, the Neumann model and the Neumann result follow as a special (Neumann economy) case from the thermodynamic based economics (irreversible economics).



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Figure 1 Lifetime as a function of interest rate

Figure 2 Production intensity as a function of time for a=1.0005


Figure 3 Production intensity as a function of time for a=1.0005


Appendix 1

Balance equations in economics

Material goods are physical (tangible) objects. As physical objects the laws of physics determine their changes. For the change of the quantity of the material goods the balance equation has to be satisfied. Let Xki be the quantity of stock i of agent k, then

Xki (t+1) = Xki (t) + Jki + Gki                                                                   (9)

where t is for time. The discrete nature of economic processes requires the discrete time approach. The length of the interval is defined by the very nature of the investigated activity. Jki is the flow of good i, that is the trade with other agents, and the trade with nature.

Jki = ĺl Jkl,i                                                                                                (10)

where Jkl,i is the flow coming from agent l to agent k, and

Jkl,i = - Jlk,i                                                                                                 (11)

Gk,i is the source/sink term, it describes the effect of production and consumption.

Conservation law of physics demands for the conservation. The quantity can change only if it is transformed to another good (production), or it is a result of transformation of other goods (bread is transformed from flour and from other input materials). The other way of change is that the goods enter to the ownership of the agent or they leave it. The “balanced” balance equation has to incorporate also the special agents (which don’t make decisions), and the free goods.

In this “balanced” description the same argument can be applied as in case of chemical reactions: there is a defined relation between the inputs and outputs, and this relation can be formulated as in the case of the stoichiometric coefficients. In production the stock change is

Gki =  bij·xi - aij·xi                                                                                     (12)

where xi is the level of the corresponding production activity of agent i. (In reality the agent ‑factory‑ has several possibilities to realize the production which lead to different A and B matrices. In the present approach we neglect that possibility.)

The balance equation written for material goods can be applied also for two other economic quantities which are not present in material form, namely for the money and for the labor. Balance for money, M has the same structure:

M(t+1) = M(t) + JM + GM                                                                        (13)

where JM is the money flow. In trade JMk = -ĺi pi Jik, and GM is the source/sink of money. GM = 0 for the normal economic agents in the present economies, it is not zero for banks, who create the money. The agents (in reality) can destroy the money, but for normal agents (non banks) positive GM is forbidden by the law and negative GM is very rare.

Labor has the peculiarity that it cannot be conserved: it is produced and used simultaneously. So it is a special service with zero stock. Nevertheless the balance equation holds. This property of labor was noticed and used already by Neumann.