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.

To Olga, Ksenia and Anastasia with love and gratitude

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author.

The monograph offered to the attention of readers is the second in the series [1, 2, 3] 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. The main point of the concept of the physical modeling is that formal approaches and procedures of theoretical physics are used to describe these economic concepts. 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.

It is also vital to the elaboration of the main paradigm and eventually the five general principles of physical economics, as well as of probabilistic economic theory. The uncertainty and probability principle holds a central position among of them. All the principles provide the necessary background to physical economics that include such theories as classical economy, probability economics, and quantum economy. The book provides a unique source for learning and understanding all the concepts and principles of physical economics, together with the quantitative methods of calculating and analyzing the many-good, many-agent market economies. Conceptually, the book can be viewed as an introduction to economics for physicists, expressed by means of the terms and language of physics. 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.

References:

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

2. Kondratenko A.V. Probabilistic Economic Theory. Novosibirsk: Nauka (Science), 2015.

3. Kondratenko A.V. Probabilistic Theory of Stock Exchanges. Novosibirsk: Nauka (Science), 2021.

PREFACE

I have the pleasure of bringing to the reader’s notice a book that expounds on the principles of physical economics, and more specifically, probabilistic economic theory. In the broad sense, physical economics is a relatively new economic discipline concerned with the study of economic phenomena from the point of view of physics. More exactly, it is through such economic investigations that we explore methods of theoretical physics previously developed in that discipline for solving formally similar problems. Depending on the statement of the economic problem under study, those can be mechanical, statistical, thermodynamic or other methods. For this reason, there are several corresponding variants of presentation of physical economics in the literature depending on the methods applied. The two books [1, 2] can be cited as an example of the presentation of physical economics from the point of view of statistical physics. It is evident that these books primarily deal with open financial markets with huge numbers of market agents, where statistical effects obviously play one of the leading roles. The key review [3] presents a phenomenological version of physical economics in terms of classical mechanics.

As far as my book is concerned, I am interested here mainly in ordinary economies with various markets of goods and commodities. In contrast to financial markets, I believe both the deterministic and probabilistic effects to play dominant roles in such market economies. From the one side, the market agent pursues, as a rule, definite aims and explores standard work methods on the markets that, to some extent, lead to determinism on the markets. From the other side, the market agents are generally forced to work under uncertain market conditions. It means that we always have to take into account in the theory that uncertainty permanently accompanies all of the market agents’ important decisions. There is no escaping the conclusion that all significant market phenomena have a probabilistic nature, too. Therefore, I study the many-agent market economic systems in this book from the point of view of classical and quantum mechanics which have been elaborated through physics to describe the deterministic and probabilistic effects in the many-particle physical systems. For this reason, I treat the term physical economics in the book in the narrow sense of this concept, from the point of view of classical and quantum mechanics of the many-particle systems. More exactly, physical economics here is a study of the formal agent-based physical models of the market economic systems. Finally, I define physical economics here primarily as the science of the agent-based physical modeling of the market economic systems, with the aid of methods and approaches worked out earlier in classical and quantum mechanics. When applying ideas of quantum mechanics to the many-agent economic systems, we inevitably obtain probabilistic economic theory which is understood in this book as most of physical economics. Note that all the physical economic models here are also agent-based ones. I think that there is great advantage to applying the agent-based approach of physical modeling because it makes it possible at the micro level, i.e., at the level of separate agents, to find a basis for explaining economic phenomena at the macro level, i.e., at the level of the whole economy. Analogously, quantum mechanics first explained the behavior of the separate electron in the deep pit. At the time, it undertook the explanation of the macro effects and calculation of macro quantities on the basis of the knowledge obtained there.

In essence, a new physical economic picture of the market world is drawn in the book. There is a huge number of formulas in it, and practically none of them are borrowed from economic literature. However, they all have their analogues in the picture of the physical world, expressed in theoretical physics or, even more accurately, in classical and quantum mechanics.

Physical economics is a proper economic discipline, since, in contrast to physics, the objects of its studies are the actions of real subjects of the economy but not the real subjects themselves. It addresses actions of real people in the real economic world, first of all; of buyers and sellers on the markets, focused on battling for their material interests and simultaneously achieving mutually advantageous cooperation. I follow the idea of classical economic theory in which the economy is simultaneously both the product and the process of human action. Moreover, in contrast to the physical world, both the structure of the economy as well as forms and methods of human action continuously and rapidly vary with time as a result of the general human evolution, as well as scientific and technical progress. Therefore, the economic laws should be derived from the study of practical human activities, but in no way by means of fitting of the known physical laws to the economic world. But the situation is reversed if we want to use theoretical methods of physics in search of the economic laws and to develop quantitative economic theories. The point is that physics has elaborated the enormous number of mathematical methods and apparatuses that describe the structure and dynamics of diverse physical systems, from the simple to the complex. And, there is nothing that would forbid us fruitfully applying these formal mathematical methods in economics.

I emphasize that the ideas, concepts and principles of classical economic theory constitute the foundation of physical economics in my understanding, and the approaches and methods of theoretical physics play here the role of the second plan. The task of these theoretical instruments is to give the adequate mathematical description of these ideas, concepts and principles. These help create the formal framework of the theory, as well as develop mathematical apparatus for describing the structure and behavior of market economies. Why is this possible? The point is that, in structure and properties, the many-agent market economic systems are quite similar to the many-particle physical systems. Take, for example, the polyatomic molecules. Indeed:

1. Markets consist of agents. Molecules consist of atoms.

2. Agents interact between themselves. Atoms interact between themselves.

3. Everything that markets do, the interacting agents do. Everything that molecules do, the interacting atoms do.

4. Dynamics of markets are determined by a principle of maximization (e.g. the trade maximization principle). Dynamics of molecules is determined by a principle of maximization (e.g. the least action principle).

5. Uncertainty and probability is an inherent important property of the market behavior. The same is valid for dynamics of molecules.

And this is still far from a complete enumeration of coincidences and analogies between the economic and physical systems.

It is widely-known that presently, advanced mathematical methods are applied in describing the dynamics of complex economic systems much less frequently than in physics. As far as the difference in the level of the penetration of formal mathematical methods into economics and physics is concerned, it is possible with reasonable caution to assert that this difference does not lie in the fact that in principle, mathematics cannot be widely used in economics by its very nature. Instead, physics has proven to be the more developed science mathematically in modern times for a variety of historical and technological reasons. This has been the situation for the past 300 years. Beautiful mathematical models have been created in physics during this time frame to describe dynamic phenomena in many-particle systems. Unfortunately, the same cannot be said for economic theory. It is obvious to me that, things being as they are, the correct conclusion for us now must not be to preserve the status quo. Nor should economists urgently develop their own unique mathematical calculations to describe economic phenomena, independent of physics. After all, why re-invent the wheel? All that is needed is to make use of some of the most salient and staggering achievements of humanity at the present time by borrowing from theoretical physics. These can be put to use for the good of the development of economic science and the global economy. This book is one of the many steps in the right direction along the proper road. Hopefully I am not wrong, although there is always that chance. There is no doubt that it will be a long road to this accomplishment, and most certainly not a fast one.

I am well aware that the very idea of using methods of theoretical physics, especially quantum mechanics, for describing economic phenomena must cause a healthy dose of skepticism from the physicists. Therefore, I emphasize that the discussion deals with the fact that only the mathematic framework of theoretical models of the respective physical systems are transferred to the physical economic models.

I incorporate into economics only the formal structural aspects of physical theories. First of all are the equations of motion for the many-particle systems, which just by themselves must not be too rigidly attached to real physical microscopic objects. Equations — they are just equations and nothing more, and if they are a beneficial descriptive tool in another science, why not make use of them? I repeat that this is just a useful mathematical object which can and should be used as a theoretical tool where it can provide benefit. For instance, in quantum mechanics wave functions and the Schrödinger equations have been successfully used for the incorporation of the uncertainty and probability principle into physics. Why, then, can we not then apply the same mathematical apparatus to the analogous uncertainty and probability principle in economic theory for purposes of mathematical description?

It is obvious that this is only an initial approximation to reality. But we are talking about modeling economic systems, and models are only models, giving only an approximate shape to the object being modeled. My physical models of economic systems also do not pretend that they are complete and precise; they can give only the approximate patterns of our market economic world, only the specific stage in our understanding of the real economic world transposed into the language of mathematics. A physical economic model is nothing other than a certain ideal, imagined construction, aimed at explaining one or more aspects of the studied phenomenon. The question is not whether it is correct or not, but whether it is useful in helping to reach a true understanding of the real economic world. Nothing more. I think that by means of this approach, some insight into the important market economy phenomena has been gained in this study.

Ten years ago I published the small book “Physical Modeling of Economic Systems: The Classical and Quantum Economies” [4]. It was my first attempt to develop an economic theory ab initio, and constructed an axiomatic basis of the theory from a limited set of first principles. The basic hallmarks of the theory that made it probabilistic and quantitative are as follows.

First. A careful, step-by-step development of the market agent-based physical economic models, where market agents play a main role in market phenomena.

Second. The complete integration of uncertainty and probability perspectives throughout the theory.

Third. A unifying, analytic framework that uses equations of motion in the formal price economic space to describe economy evolution in time.

For the last ten years, I have continually strived to advance the theory and to make it more clear and justified. In particular, for this purpose I developed the special mathematical apparatus, which is referred to in the book as probability economics. Still, I considerably advanced the theory by means of taking into consideration quantities of market goods as independent variables along with their prices. Due to this innovation, the economic price space was expanded up to the economic price-quantity space. During this time I also developed mathematical apparatus for describing the many-good, many-agent market economies. Despite the fact that achievements and expansions of theory mentioned above are very substantial, my new book carries the title “Probabilistic Economic Theory” and is, in essence, the second extended edition of my first book, in which I presented only very beginnings of the method of the agent-based physical modeling of economic systems and the basics of probabilistic economic theory.

In this book, the fundamental concepts of economic theory are exposed to critical rethinking for the purpose of answering such eternal questions of economic theory such as those regarding supply and demand, as well as market price and market force, market process and market equilibrium, invisible hand of market etc. I look at how all these concepts should be incorporated into economic theory and conveyed quantitatively in the same language in which physicists, chemists and other professionals in the so-called natural sciences present their theories, i.e., in the language of mathematics. In the book I presented maximally simplified models, in which only the most important special features and details of work of markets are described by means of maximally simplified mathematical apparatus. Let us stress here that the main aim of such basic models is only to reveal the essence of the studied phenomenon, not more. After this is accomplished, we can then develop the models further, including other, more sophisticated effects within them. This is the only true way of modeling science. Therefore, Chapters I–VIII are easily understood by first-year economics students. But the subsequent Chapters IX and X require an existing, thorough knowledge of physics, somewhere around the level of upper year physics courses. They only need have the slightest grasp of economic phenomena and laws of human action in the market economy, obtained, for example, in the course of reading the first chapters of this book. Generally, this book can be considered as an introduction into economics, written for physicists in standard physics terminology. The book, by the way, was initially taught as a set of lectures on economics for physics department students. If, after reading this book, a physics student has the impression that the presented physical economic models are quite simple and understandable, then I have solved a personal challenge. Indeed, I feel that the more complex the studied systems are and the phenomena within them, the simpler the model must be, taking into consideration only those effects which are of prime importance for describing the studied phenomena.

The book, as noted above, is the collection of lectures, each of which is called to answer one or several questions given above. The genre of lecture (or essay) is selected for the purpose of concentrating on the compact, clear presentation of physical economics. In it I have used a whole series of new ideas, concepts and notions for the economic theory, which arise from theoretical physics. I believe I have succeeded in avoiding the necessity of making numerous surveys and references, the like of which can be found in most other textbooks on economics and economic history. Therefore, references in the book are made only to those sources which were actually used for the fulfillment of studies, the development of models, and writing of the book. To provide convenience to students in lectures, figures and fragments of the text are reproduced several times in some chapters.

It should be emphasized once more that I borrowed ideas, concepts and notions from physics, many of which are completely consistent with the discoveries of the classical economic theories of the 19th century, the first of which being the subjective theory of value. This concerns first and foremost such important milestones in the development of classical economic theory as [5]:

1. Regularity in the sequence of market economic phenomena.

2. Exclusive and dominant roles of market agents in the market phenomena.

3. Uncertainty of the future and the probabilistic nature of all market agents’ decisions.

4. Social cooperation of market agents etc.

All these aspects of human economic activities have an exceptionally important effect on market processes and determine the course of economic development. But any formal methods of providing an adequate formal description of these economic phenomena and processes at a strict mathematical level in economic theory, until now, have been absent. Necessity and expediency of borrowing by economics from physics is substantiated by the fact that theoretical physics already developed sophisticated mathematical methods to incorporate analogous concepts into formal physical models. The method of equations of motion was first, and uses systems of differential equations of the 1st and 2nd orders, but economic theory still did not.

I would like to stress here that this creative process of the transfer of the formal methods of physics into classical economic theory presents the main point of the concept of the agent-based physical modeling of the economic systems and physical economics as a whole. One can say that, in essence, physical economics is first and foremost the mathematical apparatus of classical economic theory at the contemporary level of its development. This mathematical apparatus is borrowed from theoretical physics and has therefore practically nothing to do with the mathematical apparatus of neoclassical economics.

In order not to overload the text of the book by the descriptions of the known concepts of classical economic theory upon which I rested in this investigation, I make use of complete quotations from the fundamental monograph of Ludwig von Mises [5] as epigraphs to each part and chapter of the book, with two exceptions. This method allowed me to avoid the mixing of completely different styles of the presentation in the book, which could hinder the perception of the text by readers. This is very important for me, since I have attempted to not to disappoint the readers, and to convince them of how it is fruitful to make use of achievements in theoretical physics for the development of economics. The point is not in the book’s detail, but rather in its broader concept of agent-based physical modeling of economic systems, which, in my view, has enormous potential. Here lies, I think, a new and enormous field for investigation, in which an abundant harvest will be gathered for many decades yet to come. I hope that the readers will obtain a certain benefit from the acquaintance with this new physical economic perspective for economic theory.

References

1. Rosario N. Mantegna, Eugene H. Stanley. An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, 1999.

2. Peter Richmond, Jürgen Mimkes, and Stefan Hutzler. Econophysics and Physical Economics. Oxford University Press, 2013.

3. D.S. Chernavsky, N.I. Starkov, A.V. Shcherbakov. On some problems of physical economics. UFN, Vol. 172, N 9, pp. 1046–1066, 2002.

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

5. Ludwig von Mises. Human Action. A Treatise on Economics. Yale University, 1949.

Anatoly Kondratenko

Novosibirsk, 2015