We construct a continuous-time, New-Monetarist economy that displays an endogenous, non-degenerate distribution of money holdings. Properties of equilibria are obtained analytically and equilibria are solved in closed form in a variety of cases. Lump-sum transfers financed with money creation are welfare-enhancing when labor productivity is low whereas regressive transfers approach first best when labor productivity is high and agents are not too impatient. We introduce illiquid government bonds and draw implications for liquidity-trap equilibria. We also study transitional dynamics under quadratic preferences and the velocity of money under heterogeneous preference shocks.