The Serendipity Theorem Reconsidered: The Three-Generations Case Without Inheritance

Abstract

In his article on the optimum growth rate for population Samuelson (1975) proved within a two-generations model (individuals live and consume for two periods, but provide labor in the first period only) his famous so-called Serendipity Theorem: “At the optimum growth rate g*, private lifetime saving will just support the most golden golden-rule lifetime state”. The underlying theory of optimum growth rate for population was criticized mainly on two partly-related grounds: (i) Deardorff (1976) pointed out that Samuelson’s solution for the optimum population growth rate g*, derived only from necessary conditions for optimality, is in fact not optimal in general. In the special case in which both utility and production functions are Cobb-Douglas, Samuelson’s solution, for those parameter values for which it exists, provides a global minimum of steady-state utility. Moreover, Deardorff proved that for CES production functions with substitution elasticity (σ) greater than unity, steady-state utility can be made arbitrarily large by taking g sufficiently close to -δ (the depreciation rate). In his reply to Deardorffs note, Samuelson (1976) agreed with Deardorffs analysis and results. In addition he mentioned an argument first brought up by Mirrlees: If σ remains bounded above zero as the capital intensity k approaches infinity, for most reasonable forms of the utility function, the solution g* = -δ must be a local boundary maximum with finite utility.KeywordsInterest RatePopulation Growth RateSocial Security SystemIntergenerational TransferInternational Economic ReviewThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.