Statistical mechanics analysis of the equilibria of linear economies

TL;DR

This paper applies statistical mechanics to analyze the equilibrium properties of large linear economies, revealing phase transitions related to technological efficiency and economy stability as the ratio of industries to commodities varies.

Contribution

It introduces a replica method approach to derive macroscopic properties of large economies and identifies phase transitions driven by technological efficiency and economy size.

Findings

Existence of a phase transition at a critical ratio of industries to commodities.

Economies with inefficient technologies exhibit a transition from expansion to saturation.

Relaxing inefficiency leads to instability at high industry-to-commodity ratios.

Abstract

The optimal (`equilibrium') macroscopic properties of an economy with $N$ industries endowed with different technologies, $P$ commodities and one consumer are derived in the limit $N\to\infty$ with $n=N/P$ fixed using the replica method. When technologies are strictly inefficient, a phase transition occurs upon increasing $n$. For low $n$, the system is in an expanding phase characterized by the existence of many profitable opportunities for new technologies. For high $n$, technologies roughly saturate the possible productions and the economy becomes strongly selective with respect to innovations. The phase transition and other significant features of the model are discussed in detail. When the inefficiency assumption is relaxed, the economy becomes unstable at high $n$.