New Derivation of Conservation Laws for Optimal Control Problem and its Application to Economic Growth Models

Abstract

Solow[14] explored, with some models, the inter generational problem of optimal capital accumulation by straightforward application of the max-min principle. Hartwick introduced and investigated the savings-investment rule (so-called Hartwick’s rule) in the problem, first for a model with only one exhaustible resource[2] and then with many exhaustible resources [3]. He showed that Hartwick’s rule (together with Hotelling’s rule [5]) yields the intergenerational equity (constant consumption path). Sato and Kim[13] presented an interesting integral variational model with single exhaustible resource which gives rise to Hartwick’s rule under an implicitly assumed constant consumption path. Mimura, Fujiwara and Noho[10] developed the variational model with many exhaustible resources and discussed the intergenerational equity by generalizing the rules of Hartwick and Hotelling. They [1] made also this line of approach on the model with single exhaustible resource under implicitly assumed exponentially growing consumption. The ideas in [1], [9] and [10] were unified [11] to build up an integral variational principle for a study of optimal economic growths which can be applied effectively, e.g., to the intergenerational problem or to Tobin’s q-theory of investment [17].