A Note on Laws of Motion for Aggregate Distributions

Abstract

In the present paper, I derive the law of motion for the aggregate distribution directly from the laws of motion for the individuals’ states. By relying on concepts from measure theory, the derivation is concise and intuitive. I address random shocks both at the micro level and at the macro level. Micro-level shocks completely cancel at the aggregate level provided that a law of large numbers applies. Therefore, the law of motion for the aggregate distribution is a deterministic process in the absence of macro-level uncertainty. If there are macro-level risks, the law of motion for the aggregate distribution exhibits a stochastic component additionally. I illustrate the formalism in a model of wealth accumulation with stochastic interest rates, deriving the law of motion for the aggregate wealth distribution.

Keywords: aggregate distribution, law of motion, Kolmogorov forward equation, Fokker–Planck equation, wealth distribution, stochastic interest rates

JEL Classification: C02, C60, D30, D31, E21