A Model of Monetary Exchange with Lumpy Consumption

We construct a continuous-time model of incomplete markets where households value both deterministic consumption flows and infrequent and random opportunities of lumpy consumption. Because of lack of commitment and lack of record keeping, households cannot borrow to finance these random consumption opportunities, thereby generating a role for liquid assets. Equilibria are characterized by ex-post heterogeneity and a non-degenerate distribution of asset portfolios. We characterize the equilibria of such an economy and we show that in a variety of cases they are analytically tractable and admit closed form solutions. We provide examples where a one-time money injection has no effect on aggregate real quantities despite generating a redistribution of wealth across households and, for some preferences, an increase in social welfare. We show that inflation induces households to spend their real balances on consumption opportunities of lower value, and a deviation from price stability can be socially optimal thanks to a positive redistributional effect. The interplay between incomplete markets and the portfolio effect implies that there is also liquidity premium in illiquid assets that are not medium of exchange. Formulating the economy in delay differential equations, we also provide a novel efficient algorithm to compute the equilibrium dynamics and the distribution of real balances under general environment.