Fig. 3. Dynamics of the classical one-good, two-agent market economy in the price-quantity space, where quantity of the good traded, q, is constant. The economy is moving really in the price space.

We assume that potential V₁₂ describes the attraction between the buyer and the seller and has its minimum at the point p⁰¹². Then the solution of equations of motion describes movement or evolution of the entire economy as follows: the center of inertia of the whole system, introduced to theory by analogy with the center of inertia of the physical prototype, moves at a constant rate Ṗ, and the internal movement, i.e., of buyers and sellers relative to each other, represents an oscillation, usually anharmonic, around the point of equilibrium p⁰¹². This conclusion is trivially generalized for the case of an arbitrary number of buyers and sellers.

So we get classical economy with the following features:

1. Movement of the center of inertia at a constant rate signifies that if at some point of time a general price growth rate were Ṗ, then this growth will continue at the same rate. In other words, this type of economy implies that prices increase at a constant rate of inflation (or rate of inflation is constant).

2. Internal dynamics of economy means that economy is oscillating near the point of equilibrium. In this case, economy is found in the equilibrium state only within an insignificant period of time, just as a mechanical pendulum is, at its lowest point, in an equilibrium state for a short period of time. Moreover, rates of changes in the relative prices of sellers and buyers are maximal at the point of equilibrium, just as for the pendulum the rate of movement is also maximal at the point of equilibrium. According to our view, oscillations of economy relative to the point of equilibrium p⁰¹² represent nothing but the economy’s own business cycles, with a certain period of oscillation that is determined by solving equations of motion with specified mass m_i and potential V₁₂. These results correlate to the Walrasian cobweb model which is well known in neoclassical economics.

It is obvious that in the broad sense of the word, classical economy is the new quantitative method of describing the market economies, in which the first priority role in the establishment of market prices play the straight negotiations of buyers and sellers as to parameters of transactions. It is clear that this price formation is not intrinsic to the huge markets of contemporary economies, but is unique to the relatively small markets for the initial period of the formation of valuable market relations and corresponding markets in the distant past, when markets were small, undeveloped and by the sufficiently slow, i.e., in which the transactions were accomplished after lengthy negotiations.

In this Chapter we developed classical economies and derived the corresponding equations of motion, namely the economic Lagrange equations in the price space. Intuitively, we suppose that the applied least action principle can be treated to some extent as the market-based trade maximization principle. The relationship between these two principles becomes more clear within the framework of quantum economy (see the following Chapters). The extension of the method for the price-quantity space is straightforward therefore we will not do it here (respective formulas, figures and discussions can be found in Chapter I). Conceptually, we can regard Lagrangian as the mathematical classical representation of the market invisible hand concept. Note that, according to the institutional and environmental principle, Lagrangian include not only inter-agent interactions but also the influences of the state and other external factors on the market agents. Therefore, figuratively, we can say that the market invisible hand puts into practice simultaneously plans and decisions of both the market agents and the state, other institutions etc. As is seen from the above shown example, physical classical models or simply classical economies deserve thorough investigation, as they happen to become an efficient tool of theoretical economics. However, there are reasons to believe that quantum models where the uncertainty and probability principle is used for description of companies’ and people’s behavior in the market are more adequate physical models of real economic systems. Recall that probability concept was first introduced into economic theory by one of the founders of quantum mechanics, J. von Neumann, in the 40s of the XX-th century [4].

1. L.D. Landau, E.M. Lifshitz. Theoretical Physics, Vol. 1. Mechanics. Moscow, Fizmatlit, 2002.

2. L.D. Landau, E.M. Lifshitz. Theoretical Physics, Vol. 3. Quantum Mechanics. Nonrelativistic Theory. Moscow, Fizmatlit, 2002.

3. M. Intriligator. Mathematical Methods of Optimization and Economic Theory. Moscow, Airis-press, 2002.

4. J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1944.

In the present Chapter the notion of supply and demand functions in the market, traditional to economics, is exposed to critical rethinking from the point of view of the uncertainty and probability principle. The Stationary Probability Model in the Price Space is developed for the description of behavior of a seller and a buyer in the price space of a one-good market in an economy being in a normal stationary state. Within the framework of the model, the terms supply and demand have changed their meaning; a new definition of the seller’s supply and the buyer’s demand functions is given. These functions are probabilistic in nature and they are normalized to their total supply and demand expressed in monetary units. In other words, they are the seller’s and buyer’s probability distributions in making a purchase/sale transaction in the market for a certain sum of money, respectively. Further, with the help of the proposed additivity and multiplicativity formulas for supply and demand, the Stationary Probability Model in the Price Space is extended to economies having many goods and many agents in the price space. With this strategy the probabilistic supply and demand functions of the whole market are constructed. As a main result of the work, we have laid the groundwork for probability economics. It is defined as a new quantitative method for description, analysis, and investigation of the model as well as real economies and markets.

An old joke in a well-known economics textbook says that creating an economist is as simple as teaching a parrot to pronounce words “supply” and “demand” (S&D below). My former managerial economics lecturer shared his own humor on this subject: If one understands the theory of S&D elasticity, you‘ve got yourself a new economics professor! These jokes reflect an important role which is played in economics by the S&D concept, the formal realization of which we will call the traditional neoclassical model of S&D. Below we will give the most widespread version of the description of this model from the textbook [1]. To start with, we will see how economics defines the demand of each individual buyer [1]. It is possible to present demand in the form of a scale or a curve showing quantity q of a product that a consumer desires, is able to buy at each given prices p, and at a certain period of time. Further, the radical property of demand consists of the following: at an invariance of all other parameters (ceteris paribus), reduction of price leads to the corresponding increase of the quantity demanded. And, ceteris paribus, the inverse is also true; an increase in price leads to the corresponding reduction of the quantity demanded. In short, there is an inverse relationship between the price p and the quantity q demanded. Economists call this inverse relationship the law of demand.