Probabilistic Economic Theory

Anatoly Kondratenko, 2022

The monograph offered to the attention of readers is the second in the series devoted to the development of probabilistic economic theory. This book presents principles of physical economics, new economic discipline primarily concerned in the book with the agent-based physical modeling of the market economic systems and eventually with the elaborating of probabilistic economic theory. At the heart of physical economics and probabilistic economic theory are the well-known cornerstone concepts of classical economics, in particular the subjective theory of value, such as regularity in the sequence of market phenomena and an interdependence of those, as well as key roles of individuals’ actions and social cooperation in the many-agent market processes. This monograph may seem interesting to everyone who is engaged in research in economics, finance, econophysics or physical economics, as well as to professional investors and stock traders.

PART A. The Agent-Based Physical Modeling of Market Economic Systems

“In the course of social events there prevails a regularity of phenomena to which man must adjust his actions if he wishes to succeed. It is futile to approach social facts with the attitude of a censor who approves or disapproves from the point of view of quite arbitrary standards and subjective judgments of value. One must study the laws of human action and social cooperation as the physicist studies the laws of nature. Human action and social cooperation seen as the object of a science of given relations, no longer as a normative discipline of things that ought to be”.

Ludwig von Mises. Human Action. A Treatise on Economics. Page 6

CHAPTER I. Fundamentals of the Method of Agent-Based Physical Modeling

“For a social collective has no existence and reality outside of the individual members’ actions. The life of a collective is lived in the actions of the individuals constituting its body. There is no social collective conceivable which is not operative in the actions of some individuals. The reality of a social integer consists in its directing and releasing definite actions on the part of individuals. Thus the way to a cognition of collective wholes is through an analysis of the individuals’ actions”.

Ludwig von Mises. Human Action. A Treatise on Economics. Page 43

“The whole market economy is a big exchange or market place, as it were. At any instant all those transactions take place which the parties are ready to enter into at the realizable price. New sales can be effected only when the valuations of at least one of the parties have changed”.

Ludwig von Mises. Human Action. A Treatise on Economics. Page 231

PREVIEW. What is the Main Point of the Concept of Agent-Based Physical Modeling?

The concept of agent-based physical modeling is based on taking the known, fundamental concepts of classical economic theory, and uniting and eventually converting them into probabilistic economic theory. It is described with the help of formal approaches and methods borrowed from theoretical physics, beginning with the method of equations of motion for many-particle physical systems. The role of the theoretical physics methods is only to provide the framework for physical economics and eventually for probabilistic economic theory. This theory is developed step-by-step with the creation of the more complicated models, each subsequent step building on the last. It includes and increases the number of concepts and principles of physical economics, and reflects on one or more of several fundamental features of the market economy. The first one, the cooperation-oriented agent principle, is the cornerstone of all the physical economic models which holds that all market phenomena have their origins in agents’ actions. To put it differently, since the action of the market as a whole is a result of the actions of all the market agents and nothing other, the market agents and their actions must be at the basis of the physical economic models. In other words, according to the agent principle, all market phenomena have their origins in agents’ actions.

1. The Concept of Agent-Based Physical Modeling

It is well-known that the method of conceptual modeling of economic systems has long and widely been used in economic theory. According to the new physical economic mode of thought, the main requirement for such economic models, which determines their basic predestination, is skill, with which several important basic concepts and principles must harmoniously, competently and simultaneously be incorporated into the theory. The latter is very important since, by the definition of the problem, all concepts and principles play roles that compare their significance in the economy under study.

Despite their simplicity, the first and most well-known conceptual theories of neoclassical economics, such as supply and demand (S&D below), contributed significantly to economic science. They helped economists to better understand the basic elements of the economic world, and they gave rise to graphic conceptualizations that aided the transfer of this knowledge to others, especially students. Early on, conceptual modeling had, and continues to have, great significance. This is also the case in Austrian economics, where it bears the name method of imaginable constructions and it is considered as the basic method in economic inquires. In theoretical physics, it is not acceptable to accentuate attention on the use of models, since theoretical physics itself can rightfully be considered as the conceptual mathematical modeling of physical systems. Specifically, in theoretical physics, the most advanced methods of theoretical modeling of complex systems have been developed. Moreover, there is the required inclination towards conducting the quantitative numerical, and, as precisely as possible, calculations of structure and properties of these models. The deep structural and dynamic analogies between the many-particle physical systems and the many-agent economic systems are exploited in this book to transfer concepts and analytical methods from theoretical physics to economics.

2. The Main Paradigm of Physical Economics

For the achievement of larger clarity, let us again mention the main idea in physical economics regarding the use of analogies with physics. For this purpose, we once more express the thought of the main paradigm of physical economics as follows: physical systems consist of atoms. As was well known long ago, all that the physical systems do, atoms do. Market systems consist of market agents, buyers and sellers. And it is also known that all that the markets do, the market agents do. Both the physical and market economic systems are the complex dynamic systems, whose dynamics are determined by interaction between the elements of the system and their interaction with the environment in the widest sense of this notion. In our view, the very existence of such structural and dynamic similarity gives rise to the possibility, in principle, of building formal, many-agent physical economic models by analogy with the theoretical models of the many-particle physical systems, for example, of polyatomic molecules. It is here that the physical economic model could include all basic concepts and principles which define the work of the economy. As a result, it could sufficiently, simply, and adequately describe both the main structural features and principal dynamic characteristics of the economy being modeled, a target at which this study is precisely aimed, since no model can immediately describe everything completely. The study of economic systems by means of physical modeling must be carried out gradually, step by step, incorporating into the model ever finer effects and properties in the way that theoretical physics has done over the course of theoretical studies of complex physical systems. Thus, physical economics borrows formal approaches and model structures from theoretical physics. In other words, physical economics uses the very body or framework of the theoretical models of the many-particle systems, but not the results of theoretical studies of the concrete real physical systems. Namely this is the essence of physical modeling of economic systems. In conclusion, the obvious structural and dynamic analogy of the many-agent economic systems with the many-particle physical systems is basic to the formulation of fundamentals of the method of the agent-based physical modeling of the many-agent market economic systems in the formal economic space, and eventually, of the five general principles of physical economics as well as probabilistic economic theory.

3. The Axioms and Principles of Physical Economics

Generally speaking, our attitude towards the problem of adequate quantitative description of the agent behavior in the market as well as market S&D and price formation is based on the two rather simple axioms of a very general character.

1. The Agent Identity Axiom.

All market agents are the same, only the supplies and demands they have different. The axiom says that all market agents share common properties, depending primarily on agent revenues and expenses, or more strictly, on supply and demand (S&D below) for traded goods and services. It is these agents’ S&D that mainly determine the rational economic behavior of agents on the markets, and eventually the behavior of the whole markets. It shows a possibility of building rather common and accurate models of behavior of agents in the market, and hence the total market as a whole. It sets us on the right track for the identification and examination of the common properties in the behavior of market agents that ensure appearance of the common patterns and regularities in the course of market processes. It gives us the ability to build theoretical economic models on a fairly high scientific level by using physical and mathematical methods, which is the primary goal of physical economics and economic theory in general. We are certain that only these types of common market phenomena and processes are rightly a matter for exact scientific economic enquiry. In other words, it focuses us on building economics as an exact science in the image and after the likeness of the natural sciences.

2. The Agent Distinction Axiom.

All market agents are distinguished. The second axiom works when the first axiom fails. Thus, it defines those areas and aspects of the agents’ behavior on the markets that are the subject of the studies of other sciences of more applied nature, such as marketing, behavioral science, managerial economics, psychology, policy, etc. In other words, these social sciences are concerned with the specific nuances and peculiarities in the behavior of concrete people, agents and communities in different markets and situations, etc.

Let us stress that in economics we do not study real people, but rather the real actions of these people on markets. The people can be different but market decisions and actions of these people can be the same, depending primarily on their supplies and demands on markets. It is this fact that lies in the background of the agent identity axiom.

Hence, we will sum up everything we have stated above in the form of the five general principles of physical economics and probabilistic economic theory as follows:

1. The Cooperation-Oriented Agent Principle.

The most important concept concerning markets is as follows: every market consists of market agents, buyers and sellers, all strongly interacting with each other. There are never any mysterious forces in markets. Everything that markets do, the cooperation-oriented market agents do, and therefore only the cooperation-oriented, agent-based models can provide the reasonable and justified foundation for any modern economic theory.

2. The Institutional and Environmental Principle.

Markets are never completely closed and free; all the market agents are under continuous influences and under such external institutional and environmental forces and factors as states, institutions, other markets and economies, natural and technogenic phenomena, etc. The influences, exerted by each of these forces and factors on the structure of market prices and on the market behavior, can be completely compared with the effect from inter-agent interactions. Moreover, the action of strong external institutional and environmental factors can significantly hamper the effective work of market mechanisms and even practically suppress it in a way that results in the breakdown of the market’s invisible hand concept, well-known in classical economics. Therefore, the influence of institutional, environmental and other external factors must be adequately taken into account in the models, as well as simultaneously with the inter-agent interactions.

3. The Dynamic and Evolutionary Principle.

Markets are complex dynamic systems; all the market agents are in perpetual motion in search of profitable deals with each other for the sale or purchase of goods. Buyers tend to buy as cheaply as possible, and sellers want to obtain the highest possible prices. Mathematically, we can describe this time-dependent dynamic and evolutionary market process as motion in the price — quantity economic space of market agents acting in accordance with objective economic laws. Therefore, this motion has a deterministic character to some extent. This motion can and must be approximately described with the help of equations of motion;

4. The Market-Based Trade Maximization Principle.

On relatively free markets, the buyers and sellers consciously and deliberately enter into transactions of buying and selling with each other, since they make deals only under conditions in which they obtain the portion of profit that suits each of them. It is in no way compulsory that they aspire to maximize their profit in each concluded transaction, since they understand that the transactions can only be mutually beneficial. But they do attempt to increase their profit via the conclusion of a maximally possible quantity of such mutually beneficial transactions. Thus, it is possible to assert that the market as a whole strives for the largest possible volume of trade during the specific period of time. Consequently, we can make the conclusion that market dynamics can approximately be described and even approximate equations of motion for the market agents can be derived in turn by means of applying the market-based trade maximization principle to the whole economic system (more exactly, this principle is system-based).

5. The Uncertainty and Probability Principle.

Uncertainty and probability are essential parts of human action in markets. This is caused by the nature of human reasoning, as well as the fundamental human inability to accurately predict a future state of the markets. Furthermore, market outcome is the result of the actions of multiple agents, and no market is ever completely closed and free. For these reasons, all market processes are probabilistic by nature too, and an adequate description of all the market processes needs to apply probabilistic approaches and models in the economic price — quantity space. The uncertainty law results from this principle.

We assume that, from one side, these five general principles are capable of sufficiently and adequately describing the basic structural and dynamic properties of market economic systems and the market processes within them. From other side, they can be regarded as the basic pillars of physical economics, which carry on constructing step-by-step the bodies or frameworks of our physical economic models. These principles and their substantiation will be repeatedly discussed in more detail and step-by-step in this book. Concluding, let us stress that new probabilistic economic theory has been built on the basis of these principles in this book.

4. The Classical Economies

4.1. The Two-Agent Market Economies

As mentioned previously, below we will sequentially introduce into the theory the new concepts of physical modeling. They will be the building blocks in the construction of the body or framework of our models, which will also be filled step-by-step with new, concrete contents. We will start with the construction of the simplest physical economic models. In this paragraph we will create this with the use of analogies and formal methods of classical mechanics. These physical economic models will be referred to as the classical economies. Naturally in construction, we will use only first four principles, since only they have analogues in classical mechanics.

As we know, market agents are the buyers and sellers of goods and commodities, and as such are the major players in the market economy. They strongly interact with each other and with the institutions and the market’s external environment including other market economies. They continuously make decisions concerning the prices and quantities of good, and buy or sell those in the market. All the market agents’ actions govern the outcome of the market, which is the essence of the agent principle. We believe the agents to behave to a certain extent in a deterministic way, striving to achieve their definite market goals. This means that the behavior of market agents is, in turn, governed by the strict the economic laws in the market. The fact that these laws have until now been of a descriptive nature in classical economic theory, and they have not yet been expressed in a precise mathematic language, is not of key importance in this case. What is really important is that we believe all the market agents to act according to the economic laws of social cooperation that can be approximately described with the help of the market-based trade maximization principle.

Every market agent acts in the market in accordance with the rule of obtaining maximum profit, benefit, or some other criterion of optimality. In this respect, we believe the many-agent market economic systems to resemble the physical many-particle systems where all the particles interact and move in physical space. This is also in accordance with the same system-based maximization principle, such as the least action principle in classical mechanics which is applied to the whole physical system under study. The analogous situation exists in quantum mechanics (see below in the Part F).

The main drive of our research was to take the opportunity to create dynamic physical models for market economic systems. We construct these physical economic models by analogy with physics, or more precisely by analogy with theoretical models of the physical systems, consisting of formal interacting particles in formal external fields or external environments [1]. Let us stress that these particles are fictitious; they do not really exist in nature. Therefore, the physical systems mentioned above are also fictitious and they do not exist in nature either. They are indeed only imagined constructions and served simply as patterns for constructing the physical economic models. Thus, these physical economic models consist of the economic subsystem, or simply the economy or the market. It contains a certain number of buyers and sellers, as well as its institutional and external environment with certain interactions between market agents, and between the market agents and the market institutional and external environment. Moreover, according to the dynamic and evolutionary principle we assume that equations of motion, derived in physics for physical systems in the physical space, can be creatively used to construct approximate equations of motion for the corresponding physical models of economic systems in the particular formal economic spaces.

Let us briefly give the following reasons to substantiate such an ab initio approach for the one-good, one-buyer, and one-seller market economy. Let price functions p₁^D (t) and p₁^S (t) designate desired good prices of the buyer and seller, respectively, set out by the agents during the negotiations between them at a certain moment in time t. Analogously, by means of the quantity functions q₁^D (t) and q₁^S (t) we will designate the desired good quantities set out by the buyer and the seller during the negotiations in the market. Below, for brevity, we will refer to these desired values as the price and quantity quotations, which can or cannot be publicly declared by the buyer and the seller, depending on the established rules of work on the market. Note that the setting out of these quotations by the market agents is the essence of the most important market phenomenon in classical economic theory, namely the market process leading eventually to the concrete acts of choices of the market agents, being implemented by the buyer and seller through making deals (see below). Graphically, we can display these quotations as the agents’ trajectories of motion in the formal economic space as will be shown below. In real market life, these quotations are discrete functions of time, but, for simplicity, we will visualize them graphically (as well as supply and demand functions, see below) as continuous linear functions or straight lines. This approximate procedure does not lead to a loss of generality, since these functions and lines are necessary to us. They are only for the illustration of the mechanism of the market work and for the most general graphic representation of the motion of the market agents in the two formal economic spaces, corresponding to the two independent variables, price P and quantity Q. We will refer to this agent motion as market behavior, for brevity, and sometimes the evolution of the economy in time. All these terms are, in essence, synonyms in this context of the discussion. And for simplicity we will call these spaces the price space and the quantity space, respectively, as well as the united space as the price-quantity space.

By setting out desired prices and quantities this way, buyers and sellers take part in the market process and act as homo negotians (a negotiating man) in the physical modeling, aiming to maximum satisfaction in their attempts to make a profit on the market. This is the first market equilibrium price p^E₁ and quantity q^E₁ at a moment in time t₁^E at which the agents’ trajectories intersect, the deal takes place, and the interests of both the buyer and seller are optimally satisfied, taking inexplicitly into consideration the influence of the external environmental and institutional factors on the market in general. It is here that one can see similarity in the movement of the many-agent economic system in the price-quantity economic space (described by the buyer’s trajectories p₁^D (t), q₁^D (t) and seller’s trajectories p₁^S (t), q₁^S (t)) to the movement of the corresponding many-particle physical system in the physical space (described by the particles’ trajectories x_n(t)) which is also subject to a certain physical principle of maximization. In Fig. 1, we give the graphic representation of these trajectories of agents’ motion depending on the time with the help of the suitable coordinate systems of the time-price (t, P), and the time-quantity (t, Q), in the same manner as we do the construction of analogous particles’ trajectories in classical mechanics. Below we will demonstrate a substantial similarity with physics that is depicted in the upper part of Fig. 1, with, the trajectory of the motion of agents in the price space (P-space below) and, in the lower part of Fig. 1 — in the quantity space (Q-space below). In the aggregate, both pictures represent the motion of market agents in the price quantity space (PQ-space below).

This agents’ motion reflects the market process, which consists in changing continuously by the market agents their quotations. Note, we depicted in Fig. 1 a certain standard situation on the market, in which the buyer and the seller encountered deliberately at the moment of the time t₁ and began to discuss the potential transaction by a mutual exchange of information about their conditions, first of all the desired prices and the desired quantities of goods. During the negotiation, they continuously change these quotations until they agree on the final conditions of price p^E₁ and quantity q^E₁, at the moment in time t₁^E. Such a simplest market model is applicable, for example, for the imaginable island economy in which once a year, a trade of grain occurs between a farmer and a hunter. They use the American dollar, $. To illustrate, the situation is described below in Fig. 1. Note that in this and subsequent pictures we use arrows to indicate the direction of the agent’s motion during the market process.

Up to the moment of t₁, the market has been in the simple state of rest, there were no trading in it at all. At the moment of the time t₁, there appear the buyer and the seller of grain in it, which set out their initial desired prices and quantities of grain, p₁^D (t₁), p₁^S (t₁), and q₁^D (t₁), q₁^S (t₁). Points P and V in the graphs show the position of the buyer (purchaser) and seller (vendor) at the given instant of t₁. It is natural that the desires of buyer and seller do not immediately coincide, buyer wants low price, but the seller strives for the higher price. However, both desires and needs for reaching understanding and completing transaction remain, otherwise the farmer and the hunter will have the difficult next year. The process of negotiations goes on, the market process of changing by the agents their quotations continues. As a result, the positions of the market agents converge and, after all, they coincide at the moment in time of t₁^E, which corresponds to the trajectories’ intersection point E1 on the graphs.

Fig. 1. Trajectory diagram displaying dynamics of the classical two-agent market economy in the one-dimensional economic price space (above) and in the economic quantity space (below). Dimension of time t is year, dimension of the price independent variable P is $/ton, and dimension of the quantity independent variable Q is ton.

A voluntary transaction is accomplished to the mutual satisfaction. Further, the market again is immersed into the state of rest until the next harvest and its display to sale next year at the moment in time of t₂. Harvest in this season grew, therefore q₁^S(t₂)> q₁^S(t₁). In this situation, the seller is, obviously, forced to immediately set out the lower starting price, p₁^S(t₂)< p₁^S(t₁), while the buyer, seizing the opportunity, also reduced their price and increased their quantity of grain: p₁^D(t₂)< p₁^D(t₁) and q₁^D(t₂)> q₁^D(t₁). It is natural to expect in this case that the trajectories of the buyer and the seller would be slightly changed, and agreement between the buyer and the seller will be achieved with other parameters than in the previous round of trading.

Conventionally, we will describe the state of the market at every moment in time by the set of real market prices and quantities of real deals which really take place in the market. As we can see from the Fig. 1 real deals occur in the market in our case only at the moments t₁^E and t₂^E when the following market equilibrium conditions are valid (points E_i in Fig. 1):

In this formula, we used several new notions and definitions, whose meanings need explanation. Let us make these explanations in sufficient detail in view of their importance for understanding the following presentation of physical economics. First, in contemporary economic theory, the concept of supply and demand (S&D below) plays one of the central roles. Intuitively, at the qualitative descriptive level, all economists comprehend what this concept means. Complexities and readings appear only in practice with the attempts to give a mathematical treatment to these notions and to develop an adequate method of their calculation and measurement. For this purpose, the various theories contain different mathematical models of S&D that have been developed within the framework. In these theories, differing so-called S&D functions are used to formally define and quantitatively describe S&D.

In this book, we will also repeatedly encounter the various mathematical representations of this concept in different theories, which compose physical economics, namely, classical economy, probability economics, and quantum economy.

Even within the framework of one theory, it is possible to give several formal definitions of S&D functions supplementing each other. For example, within the framework of our two-agent classical economy, we can define total S&D functions as follows:

Thus, we have defined at each moment of time t the total demand function of the buyer, D₁⁰(t), and the total supply function of the seller, S₁⁰(t), as the product of their price and quantity quotations. These functions can be easily depicted in the coordinate system of time and S&D [T, S&D], as it was done in Fig. 2 displaying the so-called S&D diagram. As one would expect, the S&D functions intersect at the equilibrium point E. It is accepted in such cases to indicate that S&D are equal at the equilibrium point. We consider that it is more strictly to say that equilibrium point is that point on the diagram of the trajectories, where these trajectories intersect, i.e., where the price and quantity quotations of the buyer and the seller are equal. But that in this case S&D curves intersect is the simple consequence of their definition equality of prices and quantities at the equilibrium point.

The last observation here concerns a formula for evaluating the volume of trade in the market, MTV(t_i^E), between the buyer and the seller where they come to a mutual understanding and accomplishment of transaction at the equilibrium point E_i. It is clear that to obtain the trade volume (total value of all the transactions in this case), it is possible to simply multiply the equilibrium values of price and quantity that are derived from the above formula. The dimension of the trade volume is of course a product of the dimensions of price and quantity; in this example this is $. The same is valid for the dimensions of the total S&D, D₁⁰(t) and S₁⁰(t).

Fig. 2. S&D diagram displaying dynamics of the classical two-agent market economy in the time-S&D functions coordinate system [T, S&D], within the first time interval [t₁, t₁^E].

4.2. The Main Market Rule “Sell all — Buy at all”

Having a method to more or less evaluate the price quantitatively is always advantageous, as it helps us to somewhat predict market prices. Using the main rule of work on the market is used to this end, and this strategic rule of decision making can be briefly formulated as follows: “Sell all — Buy at all”. This main market rule indicates the following different strategies of market actions (action on the market is setting out quotations) for both the seller and the buyer. For the seller this strategy consists in striving to sell all the goods planned to sale at the maximally possible highest prices. Whereas for the buyer this strategy consists in the fact that it will expend all the money planned for the purchase of goods and try to purchase in this case as much as possible at the possible smallest price. Thus, the main market rule leads to the corresponding algorithms of the actions of agents on the markets, which are graphically represented in the form of agents’ trajectories in the pictures. The point of intersection and the respective trade volume in the market, MTV, are easily found with the help of the following mathematical formulas:

It is natural here to name D₁⁰(t₁) the total demand of buyer at the initial moment of trading. The sense of this quantity is in the fact that this is quantity of resources, planned for the purchase of goods, expressed in the money, although the dimension of this demand is the dimension of money price ($/ton) multiplied by the dimension of quantity (ton). In our case, this is $. We emphasize that, over the course of development of quantitative theory, this is very important in order to draw attention to the dimension of the used quantities and parameters, and to the normalization of the applied functions (see below).

By analogy with classical mechanics, we can treat these prices and quantity functions as the trajectories of movement of the market agents in the two-dimensional economic PQ-space as it was displayed in Fig. 3.

In principle, this representation gives nothing new in comparison with Figs. 1 and 2. Nevertheless, there is one interesting nuance here, in which the similarity of this diagram can be compared to the traditional picture in the conceptual neoclassical model of S&D. We will examine this question below. But let us now focus attention on the following nuances in the picture in Fig. 3. First, it is clearly shown by the arrows, that the buyer and the seller seemingly move towards each other on the price, with the seller reducing it, and the buyer, on the contrary, increasing it. From this, we can reflect on the illustration of normal market negotiation processes. Secondly, usually the quotations of quantities are reduced during the process of negotiations both by the buyer and by the seller. Clearly, all agents want to purchase or to sell a smaller quantity of goods at the compromise market price than at the most desired, presented at the very beginning of trading.

Fig. 3. Dynamics of the classical two-agent market economy in the two-dimensional economic price-quantity space within the first time interval [t₁, t₁^E].

And now we turn from the simplest economy to a more developed economy, in which the farmer and hunter gradually switch from the discrete trade system (one trade per year) to the continuous trade system on the market. Generally speaking, negotiations are conducted continuously and transactions are accomplished continuously, depending on the needs of the buyer and the seller. This would continue for many years. Taking into account this new long-term outlook it is expedient to change somewhat the method of describing the work of the market. Namely, by quotations of a quantity of goods, it is now more convenient to represent a quantity of goods during a specific and reasonable period of time, for example year, if the discussion deals with the long-standing work of the market. In this case the dimension of a quantity would be represented by ton/year. We show in Figs. 4, 5 how it is possible to graphically represent the work of the market over a long span of time. We see that before the establishment of equilibrium at point E, transactions were of course accomplished, but probably did not bring maximum satisfaction to the participants in the market. This would induce agents to continue to search for long-term compromises in prices and quantities. After reaching equilibrium, the volume of trade reaches a maximum, and participants in the market therefore attempt to further support this equilibrium.

Fig. 4. The classical stationary and non-stationary two-agent market economies in the [T, P] and [T, Q] coordinate systems in the time interval t > t^E.

Fig. 5. The classical stationary and non-stationary two-agent market economies in the [T, S&D] — coordinate system.

Here a fork appears in the following theory: — look at Figs. 5 and 6. If quotations cease to change, then the economy converts to a stationary state in which time appears to disappear. This is especially noticeable in Fig. 6, where this sort of stationary state is described by one point, E. We will label the economies in the stationary state simply the stationary economies. But if quotations vary with time, then the economy will be named the time-dependent or simply non-stationary economies. In Figs. 5 and 6 they are represented by two lines, which emanate from the equilibrium point E. If in this case the equilibrium quantity grows, then the economy is a growing one. But if it decreases, then the economy is falling one, which clearly is represented in Fig. 5. As a rule, in such cases, the total S&D behave similarly and this can be easily seen in Fig. 5. Let us note that their dimensions in this model have also changed, now equaling $ · ton/year.

Fig. 6. The classical stationary and non-stationary two-agent market economies in the economic price-quantity space at t > t^E.

4.3. The Many-Agent Market Economies

Now we will increase the level of complexity of the classical economies by examining how it is possible to incorporate several buyers and sellers into the theory. It is understandable that each market agent will have its own trajectories in the PQ-space. In principle, they can vary greatly. There is good reason to believe that there is much similarity in the behavior of all buyers in general. The same is valid of course for all sellers. The reason is as follows. There is the intense information exchange on the market, by means of which the coordination of actions is achieved among the buyers, among the sellers, as well as among the buyers and sellers. This coordination is carried out to assist the market in reaching its maximum volume of trade, since it is precisely during the process of trading that the last point is placed in the long process of preliminary business operations: production, financing, logistics, etc. This is exactly what we would have referred to earlier as the social cooperation of the market’s agents. For example, it is natural to expect that all buyers, from one side, and sellers, from other side, behave on the market in approximately the same way, since they all are guided in their behavior on the market by one and the same main rule of work on the market: “Sell all — Buy at all”.

Hence it is possible to draw from the above discussion the following important conclusion: the trajectories of all buyers in the P-space will be close to each other; therefore, the totality of all buyers’ trajectories can be graphically represented in the form of a relatively narrow “pipe”, in which will be plotted the trajectories of all buyers. It is also possible to represent all price trajectories of the buyers by means of a single averaged trajectory, p^D(t), which we will do below. We will do the same for the sellers, and their single averaged price trajectory we will designate as p^S(t).

We have a completely different situation with the quantity trajectories, since each market agent can have the very different quantities, bearing in mind the fact that the behavior of the buyers’ (sellers’) curves can be relatively similar to each other. Nevertheless, we can establish some regularities in the behavior of the whole market, being guided by common sense and the logical method. Since the quotations of quantities are real in the classical models, we can add them in order to obtain the quantity quotations of the whole market, q^D(t) и q^S(t). However one should do this separately for the buyers and sellers as follows:

where summing up of quantity quotations is executed formally for the market, which consists of N buyers and M sellers. In this case we understand that for the whole market we can draw all the same pictures as displayed in Figs. 1–6 for the two-agent market. Thus, for instance, we can represent the dynamics of our many-agent market by the help of the following pictures in Fig. 7. In it, the dynamics of many-agent market are depicted at the moment of equilibrium (curves q^D(p^D) and q^S(p^S)), as well as dynamics of the stationary economy (point E) and dynamics of the non-stationary growing and falling economies.

Fig. 7. Dynamics of the many-agent market economy in the price-quantity space. q^D(p^D) and q^S(p^S) are quantity trajectories reflecting dynamics of market agents’ quotations in time up to the moment of establishment of the equilibrium and making transactions at the equilibrium price.

4.4. The Classical Economies versus Neoclassical Economies

Let us call attention to the fact that, in Figs. 3 and 6, the quotation curve of the buyer, q₁^D (p^D), has negative slope, and the slope of the quotation curve of the seller, q₁^S (p^S), is positive. This reflects the natural desire of the buyer to purchase more at the lower price, as far as possible, and the natural desire of the seller to sell more at the higher price, as far as possible. Specifically, it is here we reveal the visual similarity of the classical economies to the known neoclassical model of S&D. But the visual similarity of picture in Figs. 3, 6 with the corresponding famous neoclassical picture in the form of two intersected lines of S&D is only formal; economic content in them is entirely different. In classical economies, this is a graphic representation of the real market process (which really occurs on the market at a given instant), while in the neoclassical economies, this picture expresses the planned actions of market agents on the market in the future. Note that in neoclassical economics it is namely the curves q^S(p^S) and q^D(p^D) that are called S&D functions. The economic content of these S&D functions can be roughly expressed thus: “the market is by itself, I am by myself”. If one price is on the market, then I purchase (or I sell) one quantity, and if it is another, then will I purchase (or I will sell) another quantity, and so forth. Thus, the market process is completely ignored in the neoclassical economic model. But we know that, in real market life, all market agents participate continuously in the market process, permanently changing price and quantity quotations, since each has made transaction price changes on the market. We will discuss neoclassical models in more detail in other chapters of the book. However, we will now develop an artificial classical economy that will be as similar to the neoclassical model as possible.

We will call this model the quasi-market economy with the “visible hand of the market” in order to distinguish it from the economies having self-organizing markets, or economies that exhibit the “invisible hand of market”. In the quasi-market economy, there is a definite chief (very strict and all-seeing by definition) of the market (visible hand of the market), to whom all agents of the market for the planned period, let us say a year, must pass very detailed and reliable plans with respect to purchase and sale of goods. These plans are compiled by agents and are given to the chief in the form of tables, which are formed according to the rule stated above. If just such a price is found on the market, then I will sell a particular volume; if it is another price, then I will purchase (or sell) another, specific volume, and so forth. These agent tables are represented in Fig. 8 for simplicity in the form of continuous straight lines, a factor which does not decrease their generality in this case.

Let us note that each agent passes its plan to the chief in the form of table, and chief itself unites data of these plans and presents them in the easy-to-use shape of the two straight lines in one picture. Common sense tells us that it is most profitable for the buyer to purchase more at the minimum price. But in this scenario the seller would want to sell less. The opposite would be true for both buyer and seller at the maximum price. Graphically, this is reflected in the fact that when point P₁ is higher than point V₁, and the point V₂ higher than point P₂, the consequence is that the slope of the curve of the buyer is be negative, and the slope of the curve of the seller, positive. It is obvious that these two straight lines will compulsorily be crossed at the point E (p^E, q^E), where the prices and quantities of the buyer and seller coincide. Next, the chief considers that these prices and quantities reflect certain equilibrium in the market, he or she calls the equilibrium price and quantity and declares that these values of price and quantity are set for the market year. Market process is, in this case, further completely eliminated from market life, in that the decisions of the market's chief completely substitutes it during the next year. We call this model economy a quasi-market one, since plans are compiled by market agents. However, they realize them in the prices and the quantities that are essentially dictated by the market’s chief.

Fig. 8. The classical two-agent quasi-market economy in the economic price-quantity space.

We consider this quasi-market, stationary classical economy to be, in essence, the neoclassical model of S&D. The graphic representation of the neoclassical model economy is, by the way, the same Fig. 8, since plans in the neoclassical theory are drawn up in precisely the same way that we described above for the quasi-market economy. But in neoclassical economics, it is considered a priori that the market itself in some manner will carry out the role, which the chief of the market fills in the quasi-market economy. But if the market process is absent and there are no actions of agents adapting to the market, then who or what will fill this role? Moreover, if the economy is in a stationary state, then all prices and quantities are already known to all participants in the market, and their plans then are graphically reduced simply to one point: E. It is here that the neoclassical model generally lacks any sense or value.

References

1. A.V. Kondratenko. Physical Modeling of Economic Systems. Classical and Quantum Economies. Nauka, Novosibirsk, 2005.

CHAPTER II. The Constructive Design of the Agent-Based Physical Economic Models

“The specific method of economics is the method of imaginary constructions… Everyone who wants to express an opinion about the problems commonly called economic takes recourse to this method… An imaginary construction is a conceptual image of a sequence of events logically evolved from the elements of action employed in its formation. It is a product of deduction, ultimately derived from the fundamental category of action, the act of preferring and setting aside. In designing such an imaginary construction the economist is not concerned with the question of whether or not it depicts the conditions of reality which he wants to analyze. Nor does he bother about the question of whether or not such a system as his imaginary construction posits could be conceived as really existent and in operation. Even imaginary constructions which are inconceivable, self-contradictory, or unrealizable can render useful, even indispensable services in the comprehension of reality, provided the economist knows how to use them properly. The method of imaginary constructions is justified by its success. Praxeology cannot, like the natural sciences, base its teachings upon laboratory experiments and sensory perception of external objects… The main formula for designing of imaginary constructions is to abstract from the operation of some conditions present in actual action. Then we are in a position to grasp the hypothetical consequences of the absence of these conditions and to conceive the effects of their existence… The method of imaginary constructions is indispensable for praxeology; it is the only method of praxeological and economic inquiry”.

Ludwig von Mises. Human Action. A Treatise on Economics. Page 236

PREVIEW. What is the Physical Economic Model?

This is the conceptual mathematical dynamic model of the many-agent economic systems in the formal space of the independent variables, prices and quantities, of all the market agents. It is built through a significant analogy with the theoretical physical models of the many-particle systems in real space, but taking consistently into account basic, specific differences in the economic and physical systems. Each model is, in essence, only an imaginary construction, which by no means completely reflects genuine reality. It is, however, capable of describing one or several basic special features of structure or functioning of the market economy in sufficiently strict mathematical language. It was primarily created to provide fresh insight into these features, and, after the addition of another model feature or interaction, to understand, precisely how it influences entire end results.

1. The Basic Concept of Physical Economic Design

By stretching a point, we can say that the basic concept of design of our physical economic models is skeuomorphism. Let us explain what this concept means in our case. As we have already mentioned repeatedly in this book, when constructing physical economic models we strive to reach a formal mathematical, linguistic and even graphical similarity to their physical prototypes. Specifically, this concerns both the structure and the dynamics, as well as the language and the methods of representation of the obtained results, including graphics. We consistently follow this basic concept of design throughout the book. Let us stress another point. Our main task in the book is the construction of economic models that, in as much as possible, highly resemble or copy the known form and custom physicists’ models of many-particle systems. This facilitates understanding of the models and makes it possible to use the existing, detailed language of physics within a new economic framework. For example, the language of wave functions and probability distributions will be widely used here below, although this, of course, unavoidably leads to the appearance in the theory of a large quantity of neologisms. This may strongly hamper the reading of texts by economists, but substantially facilitates this process for specialists in the fields of natural sciences. We think that this basic design concept is quite adequate for building physics-based models of economic systems.

We begin with the requirement that the graphical scheme of the physical models must be similar to the picture of the many-particle physical systems, such as polyatomic molecules, for instance. Main elements of our physical models of economic systems are shown schematically in Fig. 1. A large sphere covers a market subsystem or economy consisting of active market subjects: buyers who have financial resources and a desire to buy goods or commodities, and sellers who have goods or commodities and a desire to sell them. They are the sellers and the buyers who form supply and demand (S&D below) in the market. Small dots inside the sphere denote buyers, and big ones denote sellers. The cross-hatched area outside the sphere represents the institutional and external environments, or more exactly, internal institutions such as the state, government, society, trade unions etc., and the external environment including other markets and economies, natural factors etc. It is evident that all elements and factors of the system influence each other; buyers compete with each other in the market for goods and sellers compete with each other for the money of buyers. Buyers and sellers interact with each other, permanently influencing each other’s behavior. Institutions and the external environment influence all the economic agents, including not only businesses but also ordinary people. In other words, all the economic agents are influenced by institutional and external environments and interact with each other.

Fig. 1. The physical model scheme of an economic system: a market consisting of the interacting buyers (small dots) and sellers (big dots) who are under the influence of the internal institutions and the external environment beyond the market (covered by the conventional imaginary sphere).

In order to develop a physical model of the economic system, it is necessary to learn to describe in an exact, mathematical way both movements (behaviors and influences) of each economic agent, i.e., buyers and sellers, the state and other institutions etc., and interactions with each other. It is the goal to derive equations of motion for market agents — the buyers and sellers — who determine the dynamics, movement, or evolution of the market system in time.

2. The Economic Multi-Dimensional Price-Quantity Space

As we already discussed above, in order to show the movement or dynamics of an economy it is necessary to introduce a formal economic space in which this movement takes place. As an example of such space we can choose a formal price space designed by the analogy with a common physical space. We choose the prices P_i of the i-th item of goods as coordinate axes: i = 1, 2,…, L, where L is the number of items or goods (the bold P will designate below all the L price coordinates). In case there is only one good, the space is one-dimensional and represented by a single line. The coordinate system for the one-dimensional space is shown in Fig. 2.

The distance between two points in one-dimensional space p' and p"can be for instance determined by the following:

If two goods are traded on the market (L = 2), the space is a plane; the coordinate system is represented in this case by two mutually perpendicular lines (see Fig. 3).

The distance between two points p' and p"can be determined as follows:

We can build the price economic space of any dimension L in the same way. In spite of its apparent simplicity, the introduction of the formal economic price space is of conceptual importance as it allows us to describe behavior of market agents in general mathematical terms. It represents realistic occurrences, as setting out their own price for goods at any moment of time t is the main function or activity of market agents. It is, in fact, the main feature or trajectory of agents’ behavior in the market. Let us stress once again that it is our main goal to learn to describe these trajectories or the distributions of price probability connected with them. It is impossible to do this in a physical space. For example, we can thoroughly describe movement or the trajectory of a seller with goods in physical space, especially if they are in a car or in a spaceship. However, this description will not supply us with any understanding of their attitude towards the given goods; nor will it explain their behavior or value estimation regarding the goods as an economic agent.

Fig. 2. The economic one-dimensional price space for the one-good market economy.

Fig. 3. The economic two-dimensional price space for the two-good market economy.

Within the problem of describing agents’ behavior in the market, the role of the good prices P as independent variables, or a coordinates P is considered here to be in many situations a unique one for market economic systems. In these cases we can study market dynamics in the economic price spaces. But market situations occur fairly often in which we need to explicitly take into account the independent good quantity variables Q (the bold Q will designate below all the L quantity coordinates) and consequently to describe economic dynamics in the economic 2×L-dimensional price-quantity spaces. In these scenarios, we can imagine that the whole economic system is located in the multi-dimensional price-quantity space as it is displayed in Fig. 4. We have already used many aspects of this idea naturally when discussing classical economics. We will address any concerns in the upcoming chapters.

Fig. 4. The graphical model of the many-good, many-agent market economy in the economic multi-dimensional price-quantity space. It is displayed schematically in the conventional rectangular multi-dimensional coordinate system [P, Q] where, as usual, bold P and Q designate the price and quantity coordinate axes for all the goods. Again, our model economy consists of the market and the institutional and external environment. The market consists of buyers (small dots) and sellers (big dots) covered by the conventional sphere. Very many people, institutions, and natural and other factors can represent the external environment (cross — hatched area behind the sphere) of the market which exerts perturbations on market agents (pictured by arrows pointing from environment to the market).

3. The Market-Based Trade Maximization Principle and the Economic Equations of Motion

As we saw above in the example of the simplest classical economies, market agents actively make trade transactions, and there are no trade deals at all out of the equilibrium state. As the inclination of market agents’ action is to make deals, we can naturally conclude that market agents and the market as a whole strive to approach an equilibrium state that can be expressed as the natural tendency of the market to reach the maximum volume of trade. This fact can serve as a guide for using the market agents’ trajectories to describe their dynamics. Moreover, this fact gives us grounds to expect that equations of motion can be derived from the market-based maximization principle, used to describe these trajectories. Specifically, the main market rule “Sell all — Buy at all” can be regarded to some extent as a verbal expression of both the tendency of the market toward the trade volume maximum, and the principal ability to describe market dynamics by means of agent trajectories as solutions to certain equations of motion.

The second reason of we have confidence in creating a successful dynamic or time-dependent theory of economic systems in the economic spaces is based on the analogous dynamic theory of physical systems in physical space. We also admit that the reasonable starting point in the study of economic systems dynamics is with equations of motion for a formal physical prototype. This is in spite of the differences between the features of the economic and physical spaces and the features of the economic and physical systems. The type of equations in the spaces of both systems will be approximately the same, though the essence of the parameters and potentials in them will be completely different. It is normal in physics that one and the same equation describes different systems. For example, the equation of motion of a harmonic oscillator describes the motion of both a simple pendulum and an electromagnetic wave. Formal similarity of the equations does not mean equality of the systems which they describe.

The discipline of physics has accumulated broad experience in calculating the physical systems of different degrees of complexity with different inter-particle interactions and interactions of particles with external environments. It makes sense to try and find a way to use these achievements in finding solutions to economic problems. Should any of these attempts prove to be successful, it would establish the opportunity to do numerical research on the influence that both internal and external factors exert on the behaviours of each market agent, as well as the entire economic system’s activity. This process would be done with the help of computer calculations done on the physical economic models. Theoretical economics will have acquired the most powerful research device, the opportunities of which could only be compared to the result of the discovery and exploration of equations of motion for physical systems.

The next step in developing a physical model after selecting an appropriate economic space, is the selection of a function that will assist us in describing the dynamics of an economy, such as the movement of buyers and sellers in the price space. Trajectories in coordinate physical space x(t) (classical mechanics), wave functions ψ or distributions of probabilities ψ ² (quantum mechanics), Green’s functions G and S-matrices (in quantum physics), etc. are used as such functions in physics. We started above with an attempt to develop the model using trajectories in the price space p(t) by analogy with the use of trajectories x(t) of point-like particles used in classical mechanics. Below, this model is referred to as a classical model or simply, a classical economy. Below, we will use the term classical economy in the broad sense for designating the branch of physical modeling of many-agent economic systems with the help of methods of classical mechanics of many-particle systems. It is important to realize that each selection gives rise to its own equations of motion and, therefore, to different physical economic models. For example, if we select from these trajectory variants, then we obtain the economic Lagrange equations of motion and, therefore, the classical economies as the physical economic models. The discussion will deal with these models in detail in Chapter III. If we select wave functions, then we obtain at the output the economic Schrödinger equations of motion and, therefore, quantum economies (see Chapters IX and X). Without going into details here, let us say that both the Lagrange and Schrödinger equations appear as the result of applying the principles of maximization to the whole economic system. This is analogous to the maximization principles, which are explored in physics in obtaining the Lagrange and Schrödinger equations, respectively.

Strictly speaking, all these principles of maximization, both in economics and physics, are in essence a set of hypotheses. Their validity or effectiveness can be confirmed only via practical calculations and comparison of their results with the respective known laws and phenomena, as well as with the relevant big empirical data. But intuition suggests that this way of developing economic theory is most optimum at the present time. Since it is presently not known how to derive equations of motion in economics, borrowing existing theoretical structural models from physics is helpful. Since analogies can be drawn between the spaces and features of both physics and economics, we can use skeuomorphism and transfer the design models from the one discipline to the other.

We understand that in principle, equations of motion for economics can be derived with the aid of the market-based trade maximization principle. To be honest, we do not fully understand how this exactly works. According to some indirect signs, we can only surmise that the market-based trade maximization principle and maximization principles borrowed from physics, work in one direction. We will examine this more specifically in Chapter VIII.

References

1. A.V. Kondratenko. Physical Modeling of Economic Systems. Classical and Quantum Economies. Nauka (Science), Novosibirsk, 2005.