Sustainability, Hamiltonian Value, and NNP in a Cake-Eating Economy

Abstract

This paper sheds further light on the relationships between the current-value Hamiltonian, net national product (NNP), and sustainability condition in the context of a cake-eating economy. It shows that the usual interpretation of the current-value Hamiltonian as the maximum sustainable consumption/utility level is not correct and that a general condition for sustainability in this sense is the stationarity of the current-value Hamiltonian. It also derives an explicit relationship between this condition and the condition for sustainability in the sense of keeping the value of wealth intact. It then finds that while a cake-eating economy can never be sustainable in the former sense, it will be sustainable in the latter sense provided the preferences are presented by a logarithmic utility function. Further, the paper shows that, contrary to common belief, NNP is not, in general, equal to the current-value Hamiltonian, except for the highly special case of a linear utility function. It argues that, even when correctly adjusted to account for natural capital depreciation, NNP can still be a notoriously misleading indicator of true level of well-being because, depending on the degree of society's aversion to intergenerational inequality, it can underestimate or overestimate the current-value Hamiltonian along an optimal policy.