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...

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...