Optimal Savings and the Value of Population
Proceedings of The National Academy of Sciences, Vol. 104, No. 47, pp.18421-18426, November 20, 2007
36 Pages Posted: 15 Jan 2008 Last revised: 12 Feb 2008
Date Written: April 2, 2007
We study a model of economic growth in which an exogenously changing population enters in the objective function under total utilitarianism and into the state dynamics as the labor input to the production function. We consider an arbitrary population growth until it reaches a critical level (resp. saturation level) at which point it starts growing exponentially (resp. it stops growing altogether). This requires population as well as capital as state variables. By letting the population variable serve as the surrogate of time, we are still able to depict the optimal path and its convergence to the long-run equilibrium on a two-dimensional phase diagram. The phase diagram consists of a transient curve that reaches the classical curve associated with a positive exponential growth at the time the population reaches the critical level. In the case of an asymptotic population saturation, we expect the transient curve to approach the equilibrium as the population approaches its saturation level. Finally, we characterize the approaches to the classical curve and to the equilibrium.
Keywords: economic growth, optimal control, phase diagram, dynamic programming, optimal savings, genuine savings
JEL Classification: O41, C61
Suggested Citation: Suggested Citation