A law of large numbers for fast price adjustment

Abstract:

The purpose of this paper is to prove a law of large numbers for certain Markov processes involving large sets of weakly interacting particles. Consider a large finite set $A$ of "particles" which move about in $m$-dimensional Euclidean space ${R^m}$. The particles interact with each other indirectly by means of an auxiliary quantity $p$ in $d$-dimensional Euclidean space ${R^d}$. At each time $t$, a particle $a \in A$ is randomly selected and randomly jumps to a new location in ${R^m}$ with a distribution depending on $p$ and its old location. At the same time, the value of $p$ changes to a new value depending on these same arguments. The parameter $p$ moves by a small amount at each time but moves fast compared to the average position of the particles. Under appropriate hypotheses on the rules of motion, we shall prove the following law of large numbers. For sufficiently large $A$, the value of $p$ will be close to its expected value with large probability, and the average position of the particles will be close to its expected value with large probability. The work was motivated by the problem of modelling the adjustment of prices in mathematical economics, where the particles $a \in A$ are agents in an exchange economy, the position of $a$ at time $t$ is the commodity vector held by agent $a$ at time $t$, and $p$ is the price vector at time $t$.