The Existence of an Optimum Savings Program

Abstract

1. THE PURPOSE of this paper is to analyze critically the nature of the solutions that have been offered to the problem of how much a nation should optimally save. Since savings in this connection is the only alternative to consumption, this is equivalent to the problem of how much a nation should optimally consume. An optimal consumption program is one that makes a certain stipulated functional in utilities /(U(c(t))) as high as possible, subject to certain restrictions on the class of admissible utility and production functions. U here is an indicator of instantaneous utility and C(t) is consumption at point t. Two approaches have been adopted in finding this optimal program: (a) to define the functions C(t) on a finite time interval (0 < t < T) which corresponds to a finite planning horizon. This makes the domain of the functions closed and bounded. Together with the assumption of continuity of the various functions, this is in principle enough to solve the problem of the optimal program over the relevant time horizon. But the solution may be crucially dependent on the length of time period T and the valuation attached to the terminal stock of capital.2 The latter is not a meaningful concept unless we try to take into account what happens beyond T. This reveals the problem essentially as extending indefinitely over t. To tackle this problem of indefinite extension in time an alternative formulation is presented: (b) the functions C(t) are now defined