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This paper presents an analytical treatment of economic systems with anarbitrary number of agents that keeps track of the systems' interactions andagents' complexity. This formalism does not seek to aggregate agents. It ratherreplaces the standard optimization approach by a probabilistic description ofboth the entire system and agents'behaviors. This is done in two distinctsteps. A first step considers an interacting system involving an arbitrarynumber of agents, where each agent's utility function is subject tounpredictable shocks. In such a setting, individual optimization problems neednot be resolved. Each agent is described by a time-dependent probabilitydistribution centered around his utility optimum. The entire system of agentsis thus defined by a composite probability depending on time, agents'interactions and forward-looking behaviors. This dynamic system is described bya path integral formalism in an abstract space-the space of the agents'actions-and is very similar to a statistical physics or quantum mechanicssystem. We show that this description, applied to the space of agents'actions,reduces to the usual optimization results in simple cases. Compared to astandard optimization, such a description markedly eases the treatment ofsystems with small number of agents. It becomes however useless for a largenumber of agents. In a second step therefore, we show that for a large numberof agents, the previous description is equivalent to a more compact descriptionin terms of field theory. This yields an analytical though approximatetreatment of the system. This field theory does not model the aggregation of amicroeconomic system in the usual sense. It rather describes an environment ofa large number of interacting agents. From this description, various phases orequilibria may be retrieved, along with individual agents' behaviors and theirinteractions with the environment. For illustrative purposes, this paperstudies a Business Cycle model with a large number of agents.

A Path Integral Approach to Business Cycle Models with Large Number of Agents