Katalin Martinás
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) pricecost and demandsupply
inequalities, (4) firstly (the generalization) of Brouwer’s fixedpoint 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 saddlepoint 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 thermodynamiclike (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. 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 inputoutput model of Leontief (1954),
and Kornai (1984). On the other hand, the neoclassical approach considers that
aspect as trivial and nonimportant. Nevertheless, they did not incorporate the
Second Law. So this thermodynamiclike 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 Neumanntype 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.
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 longterm 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 nonneoclassical 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.
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 (nonphysical) 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
nonequilibrium 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 nonequilibrium 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]).
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 X^{a} and X^{b} are values of the variables for two systems ‘a’ and ‘b’, the variable has the value X^{a}+X^{b} 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:
dX_{ki}/dt =ĺ_{kl,i} J_{kl,i}+G_{ki} (2.2)
where index
i and l identify the subsystem, X_{ik }is the quantity of the extensive parameter k, J_{il,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 selfconsistency.
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, C_{3}H_{8}
is burned in oxygen, the products are carbon dioxide and water. This can be
written as a chemical equation:
C_{3}H_{8} + O_{2} ® CO_{2} + H_{2}O (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:
C_{3}H_{8} +5 O_{2} ® 3CO_{2} + 4H_{2}O (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 i^{th} process (q_{i}). 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
2H_{2} + O_{2} ® 2 H_{2}O (2.5)
meaning
that 2 moles of H_{2} 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 n_{H2O}, quantity of oxygen and hydrogen (n_{H2} and n_{O2}) are:
dn_{H2O}= 2 q,
dn_{H2} =  2q, (2.6)
dn_{O2 }=  q,
or in
general form:
dX_{i}/dt = J_{i} + Aq
– Bq (2.7)
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.
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
unidirectionality of natural processes is formulated in the form of a dynamic
law in nonequilibrium thermodynamics. The change (flow) is proportional to the
force (difference in intensive parameters, dY_{k})
J_{i} = ĺ L_{ik }dY_{k }(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
unidirectionality 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.).
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 midnineteenth century just prior to the elaboration of the second
law of thermodynamics.” Yet preentropic physics has been basically a theory
based on mechanical considerations, without a “timearrow”, without irreversibility.
As the concepts of modern economics were developed from this preentropic
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 unidirectionality 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.
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 decisionmaking
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 noneconomic 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, J_{i}.
Productionconsumption 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, knowhow,
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, nonimportant. 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 thermodynamiclike approach is a
nonneoclassical 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 unidirectionality
of economic processes was also formulated (but without the First law). The
noloss 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 noloss rule: an economic individual never acts if that action
would result in an immediate loss. The noloss rule is postulated as a decision
rule for economic agents instead of utility/profit maximization. It is
extremely important to note that the noloss 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 noloss 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.
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 (noloss) processes,
and dZ < 0 for forbidden (lossmaking) transactions. Noloss 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 noneconomic 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 nonnegative. Normally in trade and production processes dZ
> 0 .
(ii)
Economic
welfare comprises all goods and money, or moneyequivalents (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 Zfunction 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 Zvalue
of the good i. It is measured in
welfare/quantity units. Similarly w_{M
}= ¶Z/¶M is the marginal Zvalue of
money, and v_{i} = ¶Z/¶X_{i} /¶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 = w_{M }(ĺ v_{i}dX_{i}
+ dM) (3.1)
The
expected gain (profit) in trade of a unit of good i for price p_{i}
is F = w_{M}(v_{i} – p_{i}), 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
J_{n }= L_{n} (vB – vA) (3.2)
and the
traded quantity when agent k trades
with the agent m at price p_{i} will be
J_{km,i} = L_{km,i} (v_{i} – p_{i}) (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
pairwise exchanges. The agreed price refers to supplydemand equilibrium. The
noloss 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(vp) + L_{n}(vC)C (3.4)
v= ¶Z/¶X (¶Z/¶M)^{1} =v(X) (3.5)
Price, p comes
form the solution of equation
ĺ J_{nm,i} =ĺ L_{nm,i} (v_{i}
– p_{i} )= 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.
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.
Results of
simulations show that the global properties of solutions are not sensitive to
the form of the Zfunction, although the minor details (‘fine structure’)
naturally are affected by it. We selected the logarithmic form:
Z_{n} =ĺ_{i} X_{in }log (C_{n}M_{n}/X_{in})
(4.1)
where X_{in} is the stock i of the agent n, M_{n} is the
money stock of the agent n, C_{n} is constant.
Initial
stocks were selected as:
Table1: Initial Stock vectors 


Money 
Food 
Tools 
Labor 
Agriculture 
1000 
22.98 
18.51 
14.07 
Industry 
1000 
22.05 
19.34 
14.16 
Households 
1000 
21.97 
18.86 
14.73 

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 nonzero diagonal
(ownconsumption) terms in the inputoutput matrix, as shown below in Table 34.
Table 2: Output matrix

Food 
Tools 
Labor 
Agriculture 
2 
0 
0 
Industry 
0 
2 
0 
Households 
0 
0 
2 
Table 3: Input matrix

Food 
Tools 
Labor 
Agriculture 
1 
0.14 
0.08 
Industry 
0.40 
1 
0.36 
Households 
1.83 
1 
1 
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.
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 





Money 
Food 
Tools 
Labor 


Agriculture 
0.055 
1.87 
5.05 
10.07 


Industry 
0.055 
2.04 
4.98 
9.96 


Households 
0.055 
2.08 
4.96 
9.95 





The agents
agree in price, with the rule
p_{i }= (v_{1i}+v_{2i}+v_{3i})/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 + (vp). The money of the agents
changes as M’ = M – (vp)p. The
agents calculate the new values (defined by the new stocks). They calculate the
potential welfare increase (AvBv), and
the production intensity (x= L(AvBv)). 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 nonequilibrium 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 quasiperiodic oscillations emerge with
decreasing amplitudes, finally the system finds the equilibrium path. There is
a nearexponential 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.
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 + (BA)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
ĺ_{j} b_{ij}·x_{j} =(1+l)·ĺ_{j} a_{ij}·x_{j}, i = 1, 2, Ľ , n (5.2)
For a nonequilibrium growth
ĺ_{j} b_{ij}·x_{j} ł (1+l)·ĺ_{j} a_{ij}·x_{j}, 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
ĺ_{i} p_{i}·b_{ij}  ·ĺ_{i} v_{ij}·a_{ij} > 0 for real (5.4)
ĺ_{i} p_{i}·b_{ij}  ·ĺ_{i} v_{ij}·a_{ij}
< 0 for forbidden (5.5)
where v_{ij}_{ }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
ĺ_{i} p_{i}·b_{ij}  ·ĺ_{i} p_{i}·a_{ij} > 0 for real (5.6)
ĺ_{i} p_{i}·b_{ij}  ·ĺ_{i} p_{i}·a_{ij} < 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
ĺ_{i} p_{i}·b_{ij}  (1+p)·ĺ_{i} p_{i}·a_{ij} > 0 for real (5.8)
ĺ_{i} p_{i}·b_{ij}  (1+p)·ĺ_{i} p_{i}·a_{ij} < 0 for forbidden (5.9)
That
unidirectionality 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).
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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Figures
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
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 X_{ki} be the quantity of stock i of agent k, then
X_{ki} (t+1) = X_{ki} (t) + J_{ki} + G_{ki} (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. J_{ki} is the flow of good i, that is the trade with other agents,
and the trade with nature.
J_{ki} = ĺ_{l} J_{kl,i} (10)
where J_{kl,i} is the flow coming from
agent l to agent k, and
J_{kl,i} =  J_{lk,i} (11)
G_{k,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
G_{ki} = b_{ij}·x_{i}
 a_{ij}·x_{i} (12)
where x_{i} 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) + J_{M} + G_{M} (13)
where J_{M}
is the money flow. In trade J_{Mk}
= ĺ_{i} p_{i} J_{ik}, and G_{M} is the source/sink of money. G_{M} = 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 G_{M} is forbidden by the law
and negative G_{M} 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.