Читать книгу Probabilistic Economic Theory - Anatoly Kondratenko - Страница 22

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 p_i 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 p_i for short, similar to first and second time derivatives, i.e., for velocities p_i 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.

By analogy with classical mechanics we assume that these equations result from the following principle of maximization (the principle of least action or the principle of stationary action in mechanics). Namely, the action S must have the least possible value:

Fig. 1. Graphical model of an economy in the multi-dimensional PQ-space. It is displayed schematically in the conventional rectangular multi-dimensional coordinate system [P, Q] where P and Q designate all the agent price and quantity coordinate axes, respectively. Our model economy consists of the market and the external environment. The market consists of buyers (small dots) and sellers (big dots) covered by the conventional sphere. Very many people, institutions, as well as 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 here by arrows pointing from environment to market.

The obtained (1) and (2) lead to equations of motion or Lagrange equations [1]:

Equations of motion represent a system of second-order N + M differential equations of time t for N + M unknown required trajectories p_i(t).

These equations employ as yet an unknown Lagrange function or Lagrangian L (p, ṗ, t) which is to be found on the basis of research or experimental data. We will note that Lagrange functions were used in literature to solve a number of optimization problems of management science [3]. Let us emphasize that determination of the Lagrange function is the key problem that can only be solved in practice by making the data of theoretical calculation fit the experiment. It can not be done using theoretical methods only. But what we can do quickly is to make the first obvious trial step. Here we assume that to a certain degree of approximation, the Lagrange function resembles (in appearance only!) the Lagrange function of its physical prototype, a system of N+M point material particles with certain potentials. All assumptions made here can be thoroughly analyzed later at the second stage of investigation and left unchanged or made more accurate after comparison with the experimental data. Accomplishment of this stage will naturally require great effort and expense. For now, we will accept these assumptions and consider that Lagrange functions have the same form as those of their physical prototype, but all parameters and potentials of the economic system will be chosen on the basis of economic experience, not taken from the physical prototype. We consider that by adjusting parameters and potentials to the experiment we can smooth out the negative influence of assumptions made for solutions of equations of motion obtained in this particular way.

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) m₁ and m₂ 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 V₁₂ (p₁, p₂) describes interaction between the buyer and the seller (it is unknown a priori), and potentials U₁(p₁, t) and U₂(p₂, 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^E when their trajectories p₁(t^E) and p₂(t^E) intersect: p₁(t^E) = p₂(t^E) = p^E, as it is shown in Figs. 2, 3 for the model grain market. The equilibrium value of price, p^E, 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_i and potentials V₁₂ and U_i. Let us consider a case where external potentials U_i (p_i) 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 V₁₂ depends only on the module of price differences of the buyer and the seller p₁₂=|p₂ – p₁|, 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 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.