Monetary equilibrium and the differentiability of the value function

In this study we offer a new approach to proving the differentiability of the value function, which complements and extends the literature on dynamic programming. This result is then applied to the analysis of equilibrium in the recent class of monetary economies developed in [Lagos, R., Wright, R., 2005. A unified framework for monetary theory and policy analysis. Journal of Political Economy 113, 463-484]. For this type of environments we demonstrate that the value function is differentiable and this guarantees that the marginal value of money balances is well defined.