The Thermodynamic Approach to the Market Equilibrium

The theory of economics is an example of a nontrivial case in which ontological assumptions that underlie mathematical models are mathematical models themselves or, to say more precisely, are mathematical metaphors. A basic mathematical metaphor for the classical model of the market is that of a mechanical equilibrium. The basic idea of such a model is that small deviations of the system from the point of equilibrium produce “forces” which try to return the system to the equilibrium state. (Clearly, we have in mind the stable equilibrium. Though a nonstable one is no less interesting, we will only consider the stable one.) In some very important sense, “the invisible hand” of the market in this model is equivalent to a mechanical force. The economics is considered as a dynamical system. Time stands as a key notion, and the mathematical structure of the economical models is represented by a system of differential equations. In physics, it is well known that there are other, distinct approaches to the conceptualization of the intuitive notion of equilibrium. Our construction is based on the thermodynamic notion of equilibrium. According to this concept, the system gets in the state of equilibrium not because it is being affected by “forces,” but simply because this is the most probable state of the system, consisting of numerous parts, each of which is characterized by its independent dynamics. This distinction in the mathematical description of how the system changes its state is fundamental. In terms of thermodynamic approach to equilibrium, the system, instead of evolving in time, simply changes its position in the space of macroscopic parameters, remaining on a certain surface, the surface of state, singled out by the “equation of state.” Such a metaphor of equilibrium essentially differs from the mechanical one. Time does not occur here as an internal parameter of the system, the parameter that determines its dynamics, but as an external one.