Fig. 1. The physical model scheme of an economic system: a market consisting of the interacting buyers (small dots) and sellers (big dots) who are under the influence of the internal institutions and the external environment beyond the market (covered by the conventional imaginary sphere).
In order to develop a physical model of the economic system, it is necessary to learn to describe in an exact, mathematical way both movements (behaviors and influences) of each economic agent, i.e., buyers and sellers, the state and other institutions etc., and interactions with each other. It is the goal to derive equations of motion for market agents the buyers and sellers who determine the dynamics, movement, or evolution of the market system in time.
2. The Economic Multi-Dimensional Price-Quantity Space
As we already discussed above, in order to show the movement or dynamics of an economy it is necessary to introduce a formal economic space in which this movement takes place. As an example of such space we can choose a formal price space designed by the analogy with a common physical space. We choose the prices Pi of the i-th item of goods as coordinate axes: i = 1, 2,, L, where L is the number of items or goods (the bold P will designate below all the L price coordinates). In case there is only one good, the space is one-dimensional and represented by a single line. The coordinate system for the one-dimensional space is shown in Fig. 2.
The distance between two points in one-dimensional space p' and p" can be for instance determined by the following:
If two goods are traded on the market (L = 2), the space is a plane; the coordinate system is represented in this case by two mutually perpendicular lines (see Fig. 3).
The distance between two points p' and p" can be determined as follows:
We can build the price economic space of any dimension L in the same way. In spite of its apparent simplicity, the introduction of the formal economic price space is of conceptual importance as it allows us to describe behavior of market agents in general mathematical terms. It represents realistic occurrences, as setting out their own price for goods at any moment of time t is the main function or activity of market agents. It is, in fact, the main feature or trajectory of agents behavior in the market. Let us stress once again that it is our main goal to learn to describe these trajectories or the distributions of price probability connected with them. It is impossible to do this in a physical space. For example, we can thoroughly describe movement or the trajectory of a seller with goods in physical space, especially if they are in a car or in a spaceship. However, this description will not supply us with any understanding of their attitude towards the given goods; nor will it explain their behavior or value estimation regarding the goods as an economic agent.
Fig. 2. The economic one-dimensional price space for the one-good market economy.
Fig. 3. The economic two-dimensional price space for the two-good market economy.
Within the problem of describing agents behavior in the market, the role of the good prices P as independent variables, or a coordinates P is considered here to be in many situations a unique one for market economic systems. In these cases we can study market dynamics in the economic price spaces. But market situations occur fairly often in which we need to explicitly take into account the independent good quantity variables Q (the bold Q will designate below all the L quantity coordinates) and consequently to describe economic dynamics in the economic 2×L-dimensional price-quantity spaces. In these scenarios, we can imagine that the whole economic system is located in the multi-dimensional price-quantity space as it is displayed in Fig. 4. We have already used many aspects of this idea naturally when discussing classical economics. We will address any concerns in the upcoming chapters.
Fig. 4. The graphical model of the many-good, many-agent market economy in the economic multi-dimensional price-quantity space. It is displayed schematically in the conventional rectangular multi-dimensional coordinate system [P, Q] where, as usual, bold P and Q designate the price and quantity coordinate axes for all the goods. Again, our model economy consists of the market and the institutional and external environment. The market consists of buyers (small dots) and sellers (big dots) covered by the conventional sphere. Very many people, institutions, and 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 by arrows pointing from environment to the market).
3. The Market-Based Trade Maximization Principle and the Economic Equations of Motion
As we saw above in the example of the simplest classical economies, market agents actively make trade transactions, and there are no trade deals at all out of the equilibrium state. As the inclination of market agents action is to make deals, we can naturally conclude that market agents and the market as a whole strive to approach an equilibrium state that can be expressed as the natural tendency of the market to reach the maximum volume of trade. This fact can serve as a guide for using the market agents trajectories to describe their dynamics. Moreover, this fact gives us grounds to expect that equations of motion can be derived from the market-based maximization principle, used to describe these trajectories. Specifically, the main market rule Sell all Buy at all can be regarded to some extent as a verbal expression of both the tendency of the market toward the trade volume maximum, and the principal ability to describe market dynamics by means of agent trajectories as solutions to certain equations of motion.
The second reason of we have confidence in creating a successful dynamic or time-dependent theory of economic systems in the economic spaces is based on the analogous dynamic theory of physical systems in physical space. We also admit that the reasonable starting point in the study of economic systems dynamics is with equations of motion for a formal physical prototype. This is in spite of the differences between the features of the economic and physical spaces and the features of the economic and physical systems. The type of equations in the spaces of both systems will be approximately the same, though the essence of the parameters and potentials in them will be completely different. It is normal in physics that one and the same equation describes different systems. For example, the equation of motion of a harmonic oscillator describes the motion of both a simple pendulum and an electromagnetic wave. Formal similarity of the equations does not mean equality of the systems which they describe.
The discipline of physics has accumulated broad experience in calculating the physical systems of different degrees of complexity with different inter-particle interactions and interactions of particles with external environments. It makes sense to try and find a way to use these achievements in finding solutions to economic problems. Should any of these attempts prove to be successful, it would establish the opportunity to do numerical research on the influence that both internal and external factors exert on the behaviours of each market agent, as well as the entire economic systems activity. This process would be done with the help of computer calculations done on the physical economic models. Theoretical economics will have acquired the most powerful research device, the opportunities of which could only be compared to the result of the discovery and exploration of equations of motion for physical systems.