We consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables. This class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs. Examples include infinite horizon production planning under time-varying demands and equipment replacement under technological change. We provide, under a regularity condition, conditions that are both necessary and sufficient for strong duality to hold. Moreover we show that, under these conditions, the Lagrangean function corresponding to any pair of primal and dual optimal solutions forms a linear support to the optimal value function, thus extending the shadow price interpretation of an optimal dual solution to the infinite dimensional case. We illustrate the theory through an application to production planning under time-varying demands and costs where strong duality is established.