Thermodynamic Formalism of Stochastic Equilibrium Economics
Esa Nummelin, Elja Arjas
TL;DR
The paper develops a thermodynamic formalism for stochastic equilibrium economics by placing large deviations at the core of how random equilibria relate to a priori models. It defines economic entropy $I(p)=-\log P(\exists p^*:|p^*-p|<\delta)$ and a partition function $\Lambda(p)$, then shows how the Second Law and Gibbs Conditioning Principle emerge as large-deviation statements governing posterior inferences and canonical laws. It proves precise large-deviation theorems linking observable equilibria to rate functions, and it characterizes when a price is a possible equilibrium via equivalences involving the support of total excess demand and conjugate parameters $\alpha(p)$. The formalism is illustrated with ideal random economies and Cobb-Douglas settings, and extended to conditional laws of large numbers, enabling principled posterior updates and minimum-entropy posteriors in partial-observation scenarios. The approach provides a rigorous information-theoretic lens on model evaluation and posterior inference in macroeconomics, connecting economic equilibria to statistical mechanics notions such as entropy, temperature, and canonical ensembles.
Abstract
In economics, construction of perfect models in a way that would be comparable to the standards customary in physical sciences is generally not feasible. In particular, the observed value for an economic equilibrium may deviate significantly from its model-based a priori expected value. Mathematically, the a posteriori observed equilibrium may then represent a large deviation in the sense that it falls outside the region of validity of the Central Limit Theorem. With this as the motivating starting point, we propose a new approach to the theory of stochastic economic equilibrium. Drawing on recent developments in probability theory, we argue for the relevance of the theory of large deviations in stochastic equilibrium economics. Thereby the formalism of stochastic equilibrium economics becomes analogous to that of classical statistical mechanics, as the theory of large deviations forms also the mathematical basis of statistical mechanics. In consequence, thermodynamic concepts such as entropy, partition function and canonical probability can be introduced in a natural way to stochastic equilibrium economics. We focus here on the economic analogs of two fundamental principles, the Second Law of Thermodynamics and the Gibbs Conditioning Principle.