Monetary Theory with Non-degenerate Distributions

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. Directed search implies that the buyers who differ in money holdings will optimally choose to visit sellers who offer different terms of trade. With such endogenous separation of buyers, we show that the distribution of money holdings affects individual agents' decisions not directly, but rather indirectly only through a one-dimension variable – the seller's future marginal value of money. This result drastically reduces the state space of individual agents' decisions and makes the model tractable despite that the distribution of money holdings among agents is non-degenerate. We analytically characterize the monetary equilibrium and prove existence of a unique monetary steady state, using lattice-theoretic techniques. We also show that the non-degenerate distribution of money has desirable properties. Finally, we provide a calibrated example to illustrate the quantitative features of the steady state.