A tractable monetary model under general preferences

Consider the monetary model of Lagos and Wright (JPE 2005) but with general preferences and general production. I show that preferences satisfying UXXUHH - (UXH)2 = 0 is a sufficient condition for the existence and uniqueness of monetary equilibrium with degenerate money distribution. I solve for the entire class of exact solutions to the above non-linear second order partial differential equation. This class of preferences includes ones with constant return to scale, for example, constant elasticity of substitution (CES), and ones used in many other macroeconomics literatures. I also analyze the welfare implication of monetary policy in this economy.