A statistical field approach to capital accumulation

Abstract

This paper presents a model of capital accumulation for a large number of heterogeneous producer–consumer agents in an exchange space in which interactions depend on agents’ positions. Agents in the exchange space are subject to both attractive and repulsive forces: exchanges drive agents closer, but crowd out more distant agents. The formalism used in this paper was developed earlier by the authors and is based on statistical field theory. It allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and microeconomic features of the individual level. Our results show that the dynamics of capital accumulation and the agents’ positions in the exchange space are correlated. Interactions in the exchange space induce phases within the system that depend on the relative strength of the repulsive force. When the repulsive force is strong, the system is in a phase of regulated exchanges. An initial central position both favours and fastens capital accumulation in average, and high levels of initial capital drive agents towards the centre. Yet, this phase displays mild competition and a broad-based although slow improvement in exchange terms. In this phase, random shocks can redistribute capital and initiate a virtuous circle of capital accumulation. When the repulsive force is low, a phase of deregulated exchanges emerges, in which capital distribution is less homogeneous and competition among agents harshens. Increased mobility accelerates capital accumulation for high initial capital producers, whereas low initial capital producers are now evicted from the exchange space as their prices and revenues deteriorate. Thus, a threshold effect appears. Above a certain level of initial capital, agents benefit from and remain in a central position. Below this level, they remain at the periphery of the exchange space.