We study general dynamic programming problems with continuous and discrete choices and general constraints. The value functions may have kinks arising (1) at indifference points between discrete choices and (2) at constraint boundaries. Nevertheless, we establish a general envelope theorem:...

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises...

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises...

We study general dynamic programming problems with continuous and discrete choices and general constraints. The value functions may have kinks arising (1) at indifference points between discrete choices and (2) at constraint boundaries. Nevertheless, we establish a general envelope theorem:...

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises...

Classical models of money are typically based on a competitive market without capital or credit. They then impose exogenous timing structures, market participation constraints, or cash-in-advance constraints to make money essential. We present a simple model without credit where money arises...

In many moral hazard problems, the principal evaluates the agent's performance based on signals which the agent may suppress and replace with counterfeits. This form of fraud may affect the design of optimal contracts drastically, leading to complete market failure in extreme cases. I show that...

We present an envelope theorem for establishing first-order conditions in decision problems involving continuous and discrete choices. Our theorem accommodates general dynamic programming problems, even with unbounded marginal utilities. And, unlike classical envelope theorems that focus only on...