  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.
