We study the asymptotic stability of infinite horizon concave programming problems. Turnpike theorems for this class of models generally have to assume a low level of discounting. By generalizing our precedent work we provide a one-parameter family of verifiable conditions that guarantee convergence of the optimal paths to a stationary state. We call this property theta-acyclicity. In the one-dimensional case we show that supermodulatity implies our property but not viceversa. In the multidimensional case supermodularity has no relevant implications for the asymptotic behavior of optimal paths. We apply theta-acyclicity to a pair of models which study firms' dynamic behavior as based on adjustment costs. The first is the familiar model of competitive equilibrium in an industry in the presence of adjustment costs. IN the second case firms act strategically and we study the dynamic evolution implied by the closed-loop Nash equilibria. In both instances our criteria apply and allow us to obtain stability results that are much more general than those already existing in the literature.