Optimal population and capital over time: the maximum perspective.

A well-known implication of the Solow-Swan growth model with non-overlapping generations is that if individuals do not care about the number or well-being of their descendants, the present generation will gain nothing from having children (see Phelps (1966)). The non-overlapping assumption is quite crucial for such a result. Consider instead the case where generations overlap in Samuelson-Diamond fashion (Diamond (1965)), i.e. individuals live for two periods-they work in the first and retire in the second. At present, period zero, there are, therefore, two generations alive, one of which is old and the other which is young and will survive until the end of the next period. Again, the old at time zero do not, in principle, benefit from future generations but that is no longer true for the young at time zero because their capital will be operated by the young in period one. In a competitive environment, for instance, with no childbearing costs and positive crosspartial derivatives between capital and labour, the utility of the young at time zero never decreases by adding one more member to generation 1, i.e. the set of individuals born in period one, because ceteris paribus the rate of return on capital in period one also goes up. It is clear that we must adopt some allocation rule in order to be able to be more specific about the number of children that present generations must have if they want to maximize their utility. In this paper I will examine the case where present and future generations are constrained to be " fair ", in the sense of the Rawlsian maximin, to every individual who lives or is going to live during any subinterval of the planning period (stretching from zero to the indefinite future) under the assumption that it is possible to allocate consumpcion freely among individuals subject only to the technological constraints. Consequently, we are going to be looking at a typical problem of a central planning agency when its objective is to maximize the welfare of present generations but it is constrained to abide by the dictates of the maximin principle. The present approach, with its emphasis on the welfare of the living generations, is by no means typical of the optimal population literature, most of which is instead concerned with the maximization of a sum of utilities (see, e.g. Meade (1955), Dasgupta (1969, 1974), Pitchford (1972)). This objective may very well imply sacrificing present generations' utilities in favour of the future ones. Consider, for instance, the Meade-Dasgupta maximand (see Meade (1955), Dasgupta (1969)), consisting of a discounted or undiscounted sum of utilities weighted by the number of individuals in each generation. As shown by Dasgupta (1969), it is possible to find conditions under which, in the Solow-Swan model with non-overlapping generations, it would be optimal (according to his criterion) to have positive population at every point in time, implying that present generations would have to refrain from consuming all the available capital. Therefore, in a Dasgupta optimal plan the present generation is worse off than it would be if it had no descendants and ate up all