# A Path Integral Approach to Interacting Economic systems with Multiple Heterogenous Agents

** Pierre Gosselin, Aïleen Lotz, Marc Wambst**

**Abstract: **This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems’ interactions and complexity. The formalism does not seek to aggregate agents: it rather replaces the standard optimization approach by a probabilistic description of the agents behavior. This is done in two distinct steps.

A fi
rst step considers an interaction system involving an arbitrary number of agents, where each agents utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around its utility optimum.

The whole system of agents is thus de
fined by a composite probability depending on time, agents interactions, relations of strategic dominations, agents’ information sets and expectations. This setting allows for heterogeneous agents with different utility functions, strategic domination relations, heterogeneity of information, etc.

This dynamic system is described by a path integral formalism in an abstract space – the space of the agents’ actions – and is very similar to a statistical physics or quantum mechanics system. We show that this description, applied to the space of agents’ actions, reduces to the usual optimization results in simple cases. Compared to the standard optimization, such a description markedly eases the treatment of a system with a small number of agents. It becomes however useless for a large number of agents.

In a second step therefore, we show that, for a large number of agents, the previous description is equivalent to a more compact description in terms of
field theory. This yields an analytical, although approximate, treatment of the system. This
field theory does not model an aggregation of microeconomic systems in the usual sense, but rather describes an environment of a large number of interacting agents.

From this description, various phases or equilibria may be retrieved, as well as the individual agents behaviors, along with their interaction with the environment. This environment does not necessarily have a unique or stable equilibrium and allows to reconstruct aggregate quantities without reducing the system to mere relations between aggregates.

For illustrative purposes, this paper studies several economic models with a large number of agents, some presenting various phases. These are models of consumer/producer agents facing binding constraints, business cycle models, and psycho-economic models of interacting and possibly strategic agents.

**Keywords: **path integrals, statistical field theory, phase transition, non trivial vacuum, effective action, Green function, correlation functions, business cycle, budget constraint, aggregation, forward-looking behavior, heterogeneous agents, multi-agent model, strategical advantage, interacting agents, psycho-economic models, integrated structures, emergence.

**JEL Classification:** C02,C60, E00, E1