Many economic models mix continuous and discrete choice problems, such as a consumption/savings trade-off alongside a labor participation decision. These problems have non-differentiable and non-concave value functions. We show that the value function is differentiable at optimal choices if the underlying utility function is the upper envelope of differentiable functions. Hence, we obtain first-order conditions that are necessary for optimality. We do not make any concavity assumptions.

We study an economy with a fixed cost of production of a non-storable good. This simple friction gives rise to a rich equilibrium structure. Agents avoid the fixed cost by taking vacations and, hence, money arises endogenously to support trade between workers and vacationers. We show that agents acquire and spend money in cycles of finite length. Throughout such a "money cycle,'' agents decrease their consumption, which we interpret as the hot potato effect of inflation. We give an example where money holdings do not decrease monotonically throughout the money cycle.

We present a general equilibrium model of money with bank deposits and credit. Banks have two roles: first they act as safe-keepers of agents' values, second they act as transaction operators because they are able to identify agents. We show that there exists an equilibrium where money co-exists with bank deposits although interest rates payed on deposits are positive. Further, we compare our model to the basic framework where banks only act as safe-keepers and are not allowed to issue loans.