Often it is desirable to formulate certain decision problems without specifying a cut-off date and terminal conditions (which are sometimes felt to be arbitrary). This paper examines the duality theory that goes along with the kind of open-ended convex programming models frequently encountered in mathematical economics and operations research. Under a set of general axioms, duality conditions necessary and sufficient for infinite horizon optimality are derived. The proof emphasizes the close connection between duality theory for infinite horizon convex models and dynamic programming. Dual prices with the required properties are inductively constructed in each period as supports to the state evaluation function.