Total Utility, Overlapping Generations and Optimal Population

Abstract

The necessary and sufficient conditions for the solution to an optimal population growth model with overlapping generations, using the sum of total utility per generation as the objective function are derived. The solution for a specific, steady state model and an example are presented, and compared to that for a similar model without overlapping generations. In both cases, a positive, but less than infinite, optimal growth rate is found. Next, since an additively separable individual utility function is used, the differences between a total utility per generation model and a total utility per period model, a la Lerner, are discussed. Finally, the results of the total utility model are compared to those from a model in which the discounted sum of per capita utility is the objective function. An extension to the Samuelson-Lerner Utility Paradox, concerning the optimal rate of population growth, is discussed.