A Monetary Theory with Non-Degenerate Distributions

At any given point of time in an actual economy, some individuals hold more money than other individuals do. This non-degenerate distribution of money holdings among individuals is a rationale for a range of policies designed for reallocating liquidity among individuals. However, monetary theory has often abstracted from this non-degenerate distribution for tractability reasons. In this paper, we construct a tractable search model of money with a non-degenerate distribution of money holdings. We model search as a directed process in the sense that buyers know the terms of trade before visiting particular sellers, as opposed to undirected search that has dominated the literature. In this model, the distribution of money holdings among individuals is non-degenerate. We show that this distribution affects individuals' decisions not directly, but rather indirectly only through a one-dimensional variable -- the seller's future marginal value of money. This result drastically reduces the state space of individuals' decisions and makes the model tractable. We analytically characterize a monetary equilibrium, using lattice-theoretic techniques, and prove existence of a monetary steady state. In the equilibrium, buyers follow a stylized spending pattern over time, and the money distribution has a persistent wealth effect.