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

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.