We develop a two-sector monetary model with a centralized and decentralized market. Activities in the centralized market resemble those in a standard New Keynesian economy with price rigidities. In the decentralized market agents engage in bilateral exchanges for which money is essential. The model is estimated and evaluated based on postwar U.S. data. We document its money demand properties and determine the optimal long-run inflation rate that trades off the New Keynesian distortion against the distortion caused by taxing money and hence transactions in the decentralized market. Target rates of -1% or less maximize the social welfare function we consider, which contrasts with results derived from a cashless New Keynesian model.