Investigations of long-run sustainability of joint ecological-economic systems highlight the pertinence of reconsidering "free disposal" assumptions that underpin Sraffian and von Neumann approaches to value and growth theory. We investigate joint-production time-paths for square systems A and B , defined by nonnegative activity vectors T T1 y , y + ,... for periods T, T +1, ... satisfying iteratively T T 1 y y+ B = A , etc. Necessary and sufficient for existence of indefinitely sustainable "full resource utilisation" time paths (Hicks), is the condition that U ≡ BA−1 have a non-negative left-eigenvector with associated non-negative eigenvalue. Sustainable time-paths may then be balanced growth at g > −1, or convergent oscillations, or undamped cycles. Processes are classified as "essential", "non-essential", or "unsupportable" for sustainable activity, based on considerations of eigenvector (non)-negativity and (in)decomposability of U .
