Money in Search Equilibrium, in Competitive Equilibrium, and in Competitive Search Equilibrium

We compare three pricing mechanisms for monetary economies: bargaining (search equilibrium); price taking (competitive equilibrium); and price posting (competitive search equilibrium). We do this in a framework that, in addition to considering different mechanisms, extends existing work on the microfoundations of money by allowing a general matching technology and endogenous entry. We study how the nature of equilibrium and effects of policy depend on the mechanism. Under bargaining, trades and entry are both inefficient, and inflation implies a first-order welfare loss. Under price taking, the Friedman rule solves the first inefficiency but not the second, and inflation can actually improve welfare. Under posting, the Friedman rule implies first best, and inflation reduces welfare but the effect is second order.