This note gives the conditions under which there is an interior optimum rate of population growth in a two-generations-overlapping model. These conditions imply complementarity both in production and in consumption. They also validate Samuelson's serendipity theorem.
Pestieau, P., & Michel, P. (1993). Population growth and optimality. Journal of Population Economics, 6(4), 353-362. https://doi.org/10.1007/BF00599043 (Original work published 1993)