A path integral approach to business cycle models with large number of agents

Abstract

This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems' interactions and agents' complexity. This formalism does not seek to aggregate agents. It rather replaces the standard optimization approach by a probabilistic description of both the entire system and agents' behaviors. This is done in two distinct steps.A first step considers an interacting system involving an arbitrary number of agents, where each agent's 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 his utility optimum. The entire system of agents is thus defined by a composite probability depending on time, agents' interactions and forward-looking behaviors. 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 a standard optimization, such a description markedly eases the treatment of systems with 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 though approximate treatment of the system. This field theory does not model the aggregation of a microeconomic system in the usual sense. It rather describes an environment of a large number of interacting agents. From this description, various phases or equilibria may be retrieved, along with individual agents' behaviors and their interactions with the environment.For illustrative purposes, this paper studies a Business Cycle model with a large number of agents.

A statistical field approach to capital accumulation

“…Their product will be the probability description of the system. The two first variables, K and P , are standard economic variables, and their weight will be derived from the equations of the model, as proposed in Gosselin, Lotz, Wambst (2018). The last variable X is not a strictly standard economic variable.…”

“…The form of S (Ψ) is directly derived from the probabilistic description of our model (15). Technical details about the derivation of the field action S (Ψ) are given in Gosselin, Lotz and Wambst (2017) and a detailed abstract can be found in Gosselin, Lotz and Wambst (2018). In the following, we briefly recall the main steps and Appendix 1 provides some extensions adapted to our purposes.…”

“…In fact, the second order expansion of S (Ψ) around Ψ 0 (Z, θ) yields a quadratic action that determines directly the Green functions of the system. The derivation of this point can be found in (Gosselin, Lotz, Wambst 2018).…”

“…This simplification allows to compute the transition probabilities for the system in both phases, i.e. the probabilities for agents to evolve from one initial state of K, X, and P towards a final state during a certain time-span, given the environment created by the set of N agents (see Gosselin, Lotz, Wambst 2018). To do so, we first simplify in each phase the part of the action dependent on X.…”

“…Each agent is described by a time-dependent probability distribution centered around this agent's classical optimization path. In a second step, the individual agents' description is replaced by a more compact model of field theory that replicates the properties of the system when N , the number of agents, is large (Gosselin, Lotz and Wambst 2017, 2018, and Kleinert 1989. This modeling, although approximate, is compact enough to allow an analytical treatment of the system.…”