A Statistical Field Approach to Capital Accumulation
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper presents a model of capital accumulation for a large number of heterogenous producer-consumers in an exchange space in which interactions depend on agents’ positions. Each agent is described by his production, consumption, stock of capital, as well as the position he occupies in this abstract space.
Each agent produces one differentiated good whose price is fixed by market clearing conditions. Production functions are Cobb-Douglas, and capital stocks follow the standard capital accumulation dynamic equation. Agents consume all goods but have a preference for goods produced by their closest neighbors.
Agents in the exchange space are subject both to attractive and repulsive forces. Exchanges drive agents closer, but beyond a certain level of proximity, agents will tend to crowd out more distant agents.
The present model uses a formalism based on statistical field theory developed earlier by the authors. This approach allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and complexity at the individual level.
Our results show that the dynamics of capital accumulation and agents’ position in the exchange space are correlated. Interactions in the exchange space induce several phases of the system.
A first phase appears when attractive forces are limited. In this phase, an initial central position in the exchange space favors capital accumulation in average and leads to a higher level of capital, while agents far from the center will experience a slower accumulation process. A high level of initial capital drives agents towards a central position, i.e. improve the terms of their exchanges: they experience a higher demand and higher prices for their product. As usual, high capital productivity favors capital accumulation, while higher rates of capital depreciation reduce capital stock.
In a second phase, attractive forces are predominant. The previous results remain, but an additional threshold effect appears. Even though no restriction was imposed initially on the system, two types of agents emerge, depending on their initial stock of capital. One type of agents will remain above the capital threshold and occupy and benefit from a central position. The other type will remain below the threshold, will not be able to break it and will remain at the periphery of the exchange space. In this phase, capital distribution is less homogenous than in the first phase.
Key words: Path Integrals, Statistical Field Theory, Phase Transition, Capital Accumulation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: C02, C60, E00, E1.
From Rationality to Irrationality : Dynamic Interacting Structures
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This article develops a general method to solve dynamic models of interactions between multiple strategic agents that extends the static model studied previously by the authors. It describes a general model of several interacting agents, their domination relations as well as a graph encoding their information pattern. It provides a general resolution algorithm and discusses the dynamics around the equilibrium. Our model explains apparent irrational or biased individual behaviors as the result of the actions of several goal-specific rational agents. Our main example is a three-agent model describing « the conscious », « the unconscious », and « the body ». We show that, when the unconscious strategically dominates, the equilibrium is unconscious-optimal, but body and conscious-suboptimal. In particular, the unconscious may drive the conscious towards its goals by blurring physical needs. Our results allow for a precise account of agents’ time rate preference. Myopic behavior among agents leads to oscillatory dynamics : each agent, reacting sequentially, adjusts its action to undo other agents’ previous actions. This describes cyclical and apparently inconsistent or irrational behaviors in the dual agent. This cyclicality is present when agents are forward-looking, but can be dampened depending on the conscious sensitivity to other agents’ actions.
Key words: Path Integrals, Statistical Field Theory, Phase Transition, Capital Accumulation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: C02, C60, E00, E1.