Many economic models mix continuous and discrete choices, 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.

This paper presents a general equilibrium model where money is essential and agents exchange in competitive markets. A fixed cost of production induces them to take vacations during some periods. Hence, money is saved to purchase consumption during vacations. We show that agents will choose to acquire and spend money in cycles of finite length, even though aggregates are stationary. At any given time, agents have different positions on the money cycle. Throughout the money cycle, agents decrease their consumption and decrease their sensitivity to the inflation tax.

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.