Probabilistic Economic Theory

So, according to our approach, equations of motion in classical economy are nominally identical to those in the corresponding mechanical system. However, their constants and potentials will have another essence, other dimensions and other values. A great advantage of classical economies consists of the fact that mathematical solutions of these equations, analytical or numerical, have been found for a great number of Lagrange functions with different potentials. That is why it is of great help to apply them. Allow us to turn to relatively simple classical economies.

Let us consider a case of the classical economy with a single good, a single buyer, and a single seller, where environmental influence and interaction between a buyer and a seller can be described with the help of potentials. The Lagrangian of such an economy has the following form:

In (4) m

and m

are certain unknown constant values or parameters of economic agents who are the buyer and the seller respectively. The first two members of equation (4) in classical mechanics correspond to kinetic energy, and the remaining three to potential energy. Understanding the conventional character of these notions, we will use them for economy as well. Potential V

(p

, p

) describes interaction between the buyer and the seller (it is unknown a priori), and potentials U

(p

, t) and U

(p

, t) are designed to describe environmental influence on economy. They are to be chosen with respect to experimental data according to the dynamics of the modeled economy. Lagrange equations have the following form for this type of Lagrangian:

This system of two differential second-order equations of time t represents equations of motion for a selected classical economy. According to their form they are identical to the equations of motion of the physical prototype in physical space. In the latter system (5), the second Newton’s law of classical mechanics is designated: “product of mass by acceleration equals force”. And quite another matter is that potentials can be significantly different from the corresponding potentials in the physical system. We should mention once again that these potentials are to be discovered for different economies by detailed comparison of results of computation of equations of motion of economies with experimental or research data, or in other words, with data of empirical economics. At the initial stage it is natural to try to use known forms of potentials from physical theories, and we are going to do that in future. Let us note that the purchase-sale deal or transaction in the market between the buyer and the seller will take place at the time t

when their trajectories p

(t

) and p

(t

) intersect: p

(t

) = p

(t

) = p

, as it is shown in Figs. 2, 3 for the model grain market. The equilibrium value of price, p

, is indeed then the real price of the good or commodity in the market, what we refer to as the market price of the good. See formulas, figures and discussions for classical economies in Chapter I.

It is interesting that a number of some common features of classical economy with equations of motion (5) are common for almost all constants m

and potentials V

and U

. Let us consider a case where external potentials U

(p

) do not depend on time and represent potentials of attraction with high potential walls at the origin of coordinates that prevent economy from moving towards the negative price region. Further potential V

depends only on the module of price differences of the buyer and the seller p

=|p

– p

|, namely,

Fig. 2. The trajectory diagram showing dynamics of the classical one-good, two-agent market economy in the price-time coordinate system.

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 V

describes the attraction between the buyer and the seller and has its minimum at the point p

. 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 ?, and the internal movement, i.e., of buyers and sellers relative to each other, represents an oscillation, usually anharmonic, around the point of equilibrium p

. 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 ?, 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 p

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

and potential V

. 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.

3. Conclusions

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].

References