Unique monetary equilibrium with inflation in a stationary Bewley–Aiyagari model

We prove the existence and uniqueness of a stationary monetary equilibrium in a Bewley–Aiyagari model with idiosyncratic shocks. This is an exchange economy with an infinite horizon and one consumption good, and with each agent facing idiosyncratic endowment shocks at each period; the agents may trade their endowments for the only asset, fiat money. The government increases the money supply at a constant growth rate that induces inflation in a stationary monetary equilibrium. We identify the necessary and sufficient condition for a stationary monetary equilibrium (where money has a positive value and the aggregate real balance is constant over time) to exist, and, when it exists, we show that it is unique. The argument for uniqueness is based on a new monotonicity result for the average optimal consumption.