Sustainability and Hamiltonian Value

Abstract

The relationships among the Hamiltonian, NNP, and the level of sustainable consumption/utility have been widely misunderstood. This paper dispels the misconceptions and provides further new insight into these relationships. We show generally that for autonomous dynamic optimizing economies, a necessary and sufficient condition for sustainability is the stationarity of the current-value Hamiltonian. For autonomous cases, this stationarity condition generalizes Dixit et al.'s (1980) "zero-net-aggregate-investment" rule of sustainability, which in turn generalizes Solow-Hartwick's sustainability rule. For non-autonomous cases, however, except when the net "pure time effect" is constant over time, the stationarity condition is unfulfilled. In non-autonomous cases, Weitzman's (1976) "stationary equivalence" result does not hold, and the current-value Hamiltonian will underestimate (overestimate) the true welfare level when the net "pure time effect" is positive (negative). However, for the special non-autonomous case of a time-dependent utility discount rate we obtain a condition on the discount rate function that upholds the results obtained for autonomous cases. In turn, this condition extends Michel's (1982) transversality condition for the infinite-horizon autonomous control problems to problems with time dependent discount rates.

Keywords: Sustainability, Current-value Hamiltonian, Net National Product

JEL Classification: D63, Q32, C61