Читать книгу Physical Modeling of economic systems - Anatoly Kondratenko - Страница 5

To show the movement or dynamics of economy it is necessary to introduce a space in which this movement takes place. As an example of such space we choose a commodity-and-price space, or to be more exact simply a price space created by the analogy with a common physical space, where, though, we choose prices p_i of the i-th item of commodities as coordinate axes: i = 1 , 2 , … , T, where T is the amount of items of commodities. In case there is one commodity, the space is one-dimensional, i.e. it is represented by one line; the coordinate system for one-dimensional space is shown in Fig. 3.1. The distance between two points in one-dimensional space p′ and p″ is determined as

In case there are two commodities, the space is a plane; the coordinate system is represented by two mutually perpendicular lines (see Fig. 3.2), and the distance between two points p′ and p″ is determined as

After the analogy of it we can build a space of any dimension T.

In spite of apparent simplicity, introduction of a price space is of conceptual importance as it allows to describe behavior of market agents in the mathematic language in general. This possibility really exists as setting their own price for commodities any moment of time t is the main function or activity of market agents, and it is, in fact, the main feature or trajectory of agent’s behavior in the market.

It is our main goal to learn to describe these trajectories or connected with them distributions of price probability. It is impossible to do this in a physical space, for example: we can thoroughly describe movement or trajectory of a seller with commodities in physical space, especially if he is in a car or in a spaceship, but it will not have any connection with his attitude to the given commodities and his behavior in this respect (showing his attitude to the commodities by means of his monetary estimation) as an economic agent. Within the problem of describing agents in the market, the role of price as an independent variable, or a coordinate p is considered here as a unique one for market economic systems.

Fig. 3.1. Coordinate system of the one-dimensional price space.

Fig. 3.2. Coordinate system of the two-dimensional price space.

The next step in developing a physical model after selecting a space is selection of a function with the help of which we will try to describe the dynamics of an economy, i.e. movement of buyers and sellers in the price space. Trajectories in coordinate physical space х(t) (classical mechanics), wave functions  or distributions of probabilities (quantum mechanics), Green’s functions G and S-matrices (in quantum physics), etc. are used as such functions in physics. We start with an attempt to develop the model using trajectories in the price space p(t) by analogy with the use of trajectories х(t) of pointlike bodies used in classical mechanics. Such model will in short be called below as a classical model or simply classical economy.