References
1. A.V. Kondratenko. Physical Modeling of Economic Systems. Classical and Quantum Economies. Nauka (Science), Novosibirsk, 2005.
PART B. Classical Economy
“The classical economist sought to explain the formation of prices. They were fully aware of the fact that prices are not a product of the activities of a special group of people, but the result of an interplay of all members of the market society. This was the meaning of their statement that demand and supply determine the formation of prices… They wanted to conceive the real formation of prices – not fictitious prices as they would be determined if men were acting under the sway of hypothetical conditions different from those really influencing them. The prices they try to explain and do explain – although without tracing them back to the choices of the consumers – are real market prices. The demand and supply of which they speak are real factors determined by all motives instigating men to buy or to sell. What was wrong with their theory was that they did not trace demand back to the choices of the consumers; they lacked a satisfactory theory of demand. But it was not their idea that demand as they used this concept in their dissertations was exclusively determined by “economic” motives as distinguished from “noneconomic” motives. As they restricted their theorizing to the actions of businessmen, they did not deal with the motives of the ultimate consumers. Nonetheless their theory of prices was intended as an explanation of real prices irrespective of the motives and ideas instigating the consumers”.
Ludwig von Mises. Human Action. A Treatise on Economics. Page 62
CHAPTER III. Classical Economies in the Price Space
“Prices are a market phenomenon. They are generated by the market process and are the pith of the market economy. There is no such thing as prices outside the market. Prices cannot be constructed synthetically, as it were. They are the resultant of a certain constellation of market data, of actions and reactions of the members of a market society. It is vain to meditate what prices would have been if some of their determinants had been different. Such fantastic designs are no more sensible than whimsical speculations about what the course of history would have been if Napoleon had been killed in the battle of Arcole or if Lincoln had ordered Major Anderson to withdraw from Fort Sumter.
It is no less vain to ponder on what prices ought to be”.
Ludwig von Mises. Human Action. A Treatise on Economics. Page 395
PREVIEW. What are the Economic Lagrange Equations?
Based on the belief that the dynamics of the many-agent market economies has to some extent a deterministic character, we derived the economic equations of motion by formal analogy with classical mechanics of the many-particle systems. As a result, we naturally obtained the economic Lagrange equations of motion describing dynamics of economic systems in time. It is fascinating that we can interpret Lagrangian as the mathematical classical representation of the market invisible hand concept.
1. Foundations of Classical Economy
The logic of the present Section is the following. With the understanding that, pursuing own various well-defined goals, the market agents behave to some extent in a deterministic way, in this Chapter we are going to outline the adequate and approximate equations of motion for the economy. In order to design a physical model and derive classical equations of motion for economic systems in the price space ab initio, that is with the five general principles of physical economics in mind, we first make similar approximations and assumptions needed to derive the equations of motion for physical systems. This can be found in the course of theoretical physics by Landau and Lifshitz [1, 2]. In this way we derive equations of motion for economic systems, which are similar to equations for physical systems in form, and are considered by us as an initial approximation for a physical economic model of the modelled economic system.
According to the above-stated plan of actions we could confine ourselves in this Chapter to just writing equations of motion analogous to those obtained in classical mechanics. However, we consider it useful to derive a full row of equations and to make additional comments on our actions. As we have indicated before, according to our approach to classical modeling of economic systems, every economic agent, homo negotians, acts not only rationally in his or her own interests, but also reasonably. They negotiate to reach a minimum price for the buyer and a maximum price for the seller, but also leave their counteragent a chance to gain profit from transactions or to achieve some other goals, economic or noneconomic in nature. Otherwise, transactions would take place only once, while all agents would prefer the continuation and stability of their business.
Besides, we presume that external forces are usually inclined to influence market operations positively, establishing common rules of play that favor gaining maximum profit, utility, trade volume, or something else for the whole economic system. Based on these assumptions, we have a firm belief that there are certain principles of optimization, and their effects on market agents result in certain rules of market behavior and certain equations of motion that are followed by all rational or reasonable players spontaneously or voluntarily. In our opinion, it is they who have the leading role in the market.
Concluding, let us repeat that we will derive below the economic Lagrange equations of motion by recognizing inexplicitly the following five general principles of physical economics:
1. The Cooperation-Oriented Agent Principle.
2. The Institutional and Environmental Principle.
3. The Dynamic and Evolutionary Principle.
4. The Market-Based Trade Maximization Principle.
5. The Uncertainty and Probability Principle.
It is evident that the uncertainty and probability effects begin to play significant role in classical economies only for the markets with huge numbers of agents. We do not concern ourselves with these effects within the framework of classical economy because it is much easier to study these problems within the framework of quantum economy (see next two chapters).
2. The Economic Lagrange Equations
Let us proceed to deriving equations of motion for the classical economy shown schematically in Fig. 1. We follow the same procedure as in classical mechanics [1]. To make calculations easier, we will consider here the one-good market economy, that is, only movement in one-dimensional price space with one coordinate P. Transition to a multi-dimensional case does not cause principal complications. We will consider that by the analogy with classical mechanics [1], a state of economy comprising of N buyers and M sellers and being under the influence of the environment is fully described by establishing all prices pi and their first time t derivatives (price changing rate or velocity of movement)
, where
i = 1, 2, …,
N + M. Let
p without subscripts denote the set of all prices
pi for short, similar to first and second time derivatives, i.e., for velocities
pi and accelerations
. Due to their logic or by definition, equations of motion connect prices, velocities and accelerations. In classical mechanics they are second-order differential equations of time, their solution under assigned conditions at the moment
t0, x(
t0) and
ẋ(
t0), represents the required mechanical trajectories,
x(
t). We are going to derive similar equations with the same view for our economic system in the price space.
