the influence of the external environmental and institutional factors on the market in general. It is here that one can see similarity in the movement of the many-agent economic system in the price-quantity economic space (described by the buyer’s trajectories p1D (t), q1D (t) and seller’s trajectories p1S (t), q1S (t)) to the movement of the corresponding many-particle physical system in the physical space (described by the particles’ trajectories xn(t)) which is also subject to a certain physical principle of maximization. In Fig. 1, we give the graphic representation of these trajectories of agents’ motion depending on the time with the help of the suitable coordinate systems of the time-price (t, P), and the time-quantity (t, Q), in the same manner as we do the construction of analogous particles’ trajectories in classical mechanics. Below we will demonstrate a substantial similarity with physics that is depicted in the upper part of Fig. 1, with, the trajectory of the motion of agents in the price space (P-space below) and, in the lower part of Fig. 1 – in the quantity space (Q-space below). In the aggregate, both pictures represent the motion of market agents in the price quantity space (PQ-space below).
This agents’ motion reflects the market process, which consists in changing continuously by the market agents their quotations. Note, we depicted in Fig. 1 a certain standard situation on the market, in which the buyer and the seller encountered deliberately at the moment of the time t1 and began to discuss the potential transaction by a mutual exchange of information about their conditions, first of all the desired prices and the desired quantities of goods. During the negotiation, they continuously change these quotations until they agree on the final conditions of price pE1 and quantity qE1 , at the moment in time t1E. Such a simplest market model is applicable, for example, for the imaginable island economy in which once a year, a trade of grain occurs between a farmer and a hunter. They use the American dollar, $. To illustrate, the situation is described below in Fig. 1. Note that in this and subsequent pictures we use arrows to indicate the direction of the agent’s motion during the market process.
Up to the moment of t1 , the market has been in the simple state of rest, there were no trading in it at all. At the moment of the time t1, there appear the buyer and the seller of grain in it, which set out their initial desired prices and quantities of grain, p1D (t1), p1S (t1), and q1D (t1), q1S (t1). Points P and V in the graphs show the position of the buyer (purchaser) and seller (vendor) at the given instant of t1. It is natural that the desires of buyer and seller do not immediately coincide, buyer wants low price, but the seller strives for the higher price. However, both desires and needs for reaching understanding and completing transaction remain, otherwise the farmer and the hunter will have the difficult next year. The process of negotiations goes on, the market process of changing by the agents their quotations continues. As a result, the positions of the market agents converge and, after all, they coincide at the moment in time of t1E, which corresponds to the trajectories’ intersection point E1 on the graphs.
Fig. 1. Trajectory diagram displaying dynamics of the classical two-agent market economy in the one-dimensional economic price space (above) and in the economic quantity space (below). Dimension of time t is year, dimension of the price independent variable P is $/ton, and dimension of the quantity independent variable Q is ton.
A voluntary transaction is accomplished to the mutual satisfaction. Further, the market again is immersed into the state of rest until the next harvest and its display to sale next year at the moment in time of t2. Harvest in this season grew, therefore q1S(t2)> q1S(t1). In this situation, the seller is, obviously, forced to immediately set out the lower starting price, p1S(t2)< p1S(t1), while the buyer, seizing the opportunity, also reduced their price and increased their quantity of grain: p1D(t2)< p1D(t1) and q1D(t2)> q1D(t1). It is natural to expect in this case that the trajectories of the buyer and the seller would be slightly changed, and agreement between the buyer and the seller will be achieved with other parameters than in the previous round of trading.
Conventionally, we will describe the state of the market at every moment in time by the set of real market prices and quantities of real deals which really take place in the market. As we can see from the Fig. 1 real deals occur in the market in our case only at the moments t1E and t2E when the following market equilibrium conditions are valid (points Ei in Fig. 1):
In this formula, we used several new notions and definitions, whose meanings need explanation. Let us make these explanations in sufficient detail in view of their importance for understanding the following presentation of physical economics. First, in contemporary economic theory, the concept of supply and demand (S&D below) plays one of the central roles. Intuitively, at the qualitative descriptive level, all economists comprehend what this concept means. Complexities and readings appear only in practice with the attempts to give a mathematical treatment to these notions and to develop an adequate method of their calculation and measurement. For this purpose, the various theories contain different mathematical models of S&D that have been developed within the framework. In these theories, differing so-called S&D functions are used to formally define and quantitatively describe S&D.
In this book, we will also repeatedly encounter the various mathematical representations of this concept in different theories, which compose physical economics, namely, classical economy, probability economics, and quantum economy.
Even within the framework of one theory, it is possible to give several formal definitions of S&D functions supplementing each other. For example, within the framework of our two-agent classical economy, we can define total S&D functions as follows:
Thus, we have defined at each moment of time t the total demand function of the buyer, D10(t), and the total supply function of the seller, S10(t), as the product of their price and quantity quotations. These functions can be easily depicted in the coordinate system of time and S&D [T, S&D], as it was done in Fig. 2 displaying the so-called S&D diagram. As one would expect, the S&D functions intersect at the equilibrium point E. It is accepted in such cases to indicate that S&D are equal at the equilibrium point. We consider that it is more strictly to say that equilibrium point is that point on the diagram of the trajectories, where these trajectories intersect, i.e., where the price and quantity quotations of the buyer and the seller are equal. But that in this case S&D curves intersect is the simple consequence of their definition equality of prices and quantities at the equilibrium point.
The last observation here concerns a formula for evaluating the volume of trade in the market, MTV(tiE), between the buyer and the seller where they come to a mutual understanding and accomplishment of transaction at the equilibrium point Ei. It is clear that to obtain the trade volume (total value of all the transactions in this case), it is possible to simply multiply the equilibrium values of price and quantity that are derived from the above formula. The dimension of the trade volume is of course a product of the dimensions of price and quantity; in this example this is $. The same is valid for the dimensions of the total S&D, D10(t) and S10(t).