Quantum economy

Quantum economy – this notion first occurred in an unpublished work by Anatoly V Kondratenko, "Physical Modeling of Economic Systems. Classical and Quantum Economies" that was put on the Internet in English in Novosibirsk in 2005.

Main notions

Quantum economy in the narrow sense is a quantum economic (or simply quantum) model of a finite economic system that consists of an economic subsystem (or simply economy) with a certain number of buyers and sellers (economy agents) and its external environment (institutions) with certain interactions between economy agents, and between the economy agents and institutions. All quantum models that describe finite economic systems are based on using quantum physical models of the corresponding finite physical systems consisting of point bodies. Moreover, it is assumed that equations of motion derived for physical systems in the Cartesian space are quite a good initial approximation for equations of motion of modeled economic systems in some price space. Quantum economy in the large sense is a field of science about physical modeling of finite economic systems by means of corresponding finite physical systems and studying of such economic models with the help of quantum mechanics methods. In this sense it can be called also quantum economics. Quantum economy can also be seen as a new subsection of econophysics which gives quantum physical models for finite economic systems. Whereas traditional econophysics subsections give physical models for infinite economic systems that allows to use statistical physics methods for their description.

Physical modeling of economic systems

Introduction of price space

To derive economic equations of motion a price space (more exactly, commodity-and-price space) is introduced with an analogy to Cartesian space for physical models. This price space allows to describe behavior of economy agents in the mathematic language in general because setting their own price p for the commodities in every moment of time t is the main function of economy agents in classical economy, and it is, in fact, the main feature of agents’ behavior in the market, which is bringing to the wave functions as well as demand and supply functions forming in quantum economy.

Economic dynamics functions

The next step is a function choosing that can describe the economy dynamics, namely the motion of buyers and sellers in price space. In classical economy this function is an economy agents’ trajectory of motion in price space p(t) (p is a set of all the agents’ prices on all commodities), in other words, the price dependence on time. Derivation of equation of motion here (economic Lagrange equation) is the same as in classical mechanics. In quantum economy this dynamics function is a wave function ψ (p, t) which is related with economy agents’ probabilities of sale transactions at price p and time t, namely |ψ (p, t)|²). Thus the quantum method of description of market agents’ motion leads to probabilistic mechanism of forming of demand and supply functions and market price at which sales transactions are mostly carried out in the market.In quantum economy there are two kinds of equation of motion: for time-dependent (or transitional) and stationary states.

Time-dependent economic Schrödinger equation

For time-dependent states of the economy the equation of motion is the following time-dependent economic Schrödinger equation (derivation is made by analogy with quantum mechanics): "i"α" ∂"Ψ (p, t) /"∂"t = Ĥ Ψ (p, t),

where the total wave function Ψ (p, t) describes the motion of the economy in price space, i is the usual complex variable, and α is a certain unknown constant. If in quantum mechanics this is Plank constant, in quantum economics this is just a parameter which may be different for various economies. Ĥ is a certain also unknown linear Hermitian operator of the economy that is called Hamilton operator in quantum mechanics, which in quantum economy has the meaning, that it takes into account all the interactions between buyers, sellers and institutions and determines economy dynamics. All unknown variables are to be defined from the experience, that’s why they will be different for different types of economic systems.

tationary economic Schrödinger equation

For stationary states of the economy the economic equation of motion is the following stationary economic Schrödinger equation (derivation as in quantum mechanics too):

Ĥ Ψ_n(p) = E_nΨ _n(p)

where E_n and Ψ_n(p) denote so-called energies and wave functions of stationary states n of the quantum economy, respectively, their economic sense is still also to be revealed.

Probabilistic demand and supply functions

It’s time now to determine demand and supply functions using the obtained wave functions as solutions of the given equations of motion. It is obvious, that these wave functions have probabilistic character connected with the economy agents’ behavior uncertainties. The consequence is that the demand and supply functions have also a probabilistic nature: sales transactions can be made at any price but with different probabilities. For example, for a stationary state of the economy with a single buyer and a single seller in the one-dimensional price space (one-commodity case) the probability to have a transaction f(p) is a product of the purchases d(p) (d(p)=|ψ_d(p)|²) and sales s(p) (s(p)=|Ψ_s(p)|²) probabilities: f(p)=d(p)s(p). The buyers’ demand function D(p) and the seller’s supply function S(p) can be determined as products of distribution functions d(p) and s(p) by the total demand D₀ and total supply S₀ respectively: D(p)= D₀*d(p), S(p)= S₀*s(p).

Probabilistic nature of market price

It is obvious that market price depends on interrelation between the probabilistic demand and supply functions. As the consequence of this fact, market price has probabilistic nature too. As it was already said, sales transactions can be made at any price, but with different probabilities. Consequently we can define market price as the price p_m at which maximum of the probability to make transactions f(p) is achieved.It can be easily shown that at this market price p_m elasticity of demand ED(p_m) is absolutely equal to elasticity of supply ES(p_m):

ED(p_m)|=|ES(p_m)|.

It can be easy seen that the market price p_m explicitly depends neither on the total demand D₀ nor on the total supply S₀. Thus, in quantum economy the market price p_m is only explicitly determined by behavior peculiarities of the demand and supply functions and it doesn’t coincide with the point p₀ of intersection of the demand and supply functions which is defined as market price in neoclassical economic theory

Interrelation with other theories

If compare quantum economy with other most known economic theories it can be said, that quantum economy does not contradict them. Generally speaking, quantum economy is just only method for "ab initio" or non-empirical calculations of demand and supply functions and their dynamics or evolution in time due to all interactions in economic systems. Moreover quantum economy combines most important of mainstream theories and gives fruitful theoretical and computational tools for further development of them. Really, on the one hand, quantum economy makes it possible to simultaneously consider influence exerted by the interaction of the economic agents (a major subject in neoclassical economic theory) and interaction of the government, society and other institutions with economic agents (subject of investigation by means of institutional economics) on the economy agents’ behavior, and on the other hand, to get equations of motion of the economy that describe evolution of the economy in time (paradigm of evolutionary economics). Besides, the obtained stationary economic Schrödinger equations can be solved with the help of supercomputers using modified programs for calculation of electron wave functions of molecules and solids, in other words, by means of advanced computational methods of quantum chemistry of molecules and solids.

Literature

Anatoly V Kondratenko. Physical Modeling of Economic Systems. Classical and Quantum Economies. Novosibirsk, Nauka, 2005

External links

*Kondratenko, Anatoly (2005): [http://mpra.ub.uni-muenchen.de/6123/ Physical Modeling of economic systems. Unpublished] .